Unit 13 Controllability and Observability A system is said to be controllable at timeif it is possible by means of an unconstrainedcontrol vector to transfer the system from any initial stateto any other state in a finiteinterval of time. A system is said to be observable at timeif, with the system in state,it is possible to determine this state from the observation of the output over a finite timeinterval. The concepts of the controllability and observability were introduced by Kalman. They play animportant role in the design of control systems in state space. In fact, the conditions ofcontrollability and observability may govern the existence of a complete solution of the controlsystem design problem. The solution to this problem may not exist of the system considered isnot controllable. Although most physical systems are controllable and observable,corresponding mathematical models may not possess the property of controllability andobservability.j Complete State Controllability of Continuous-Time SystemsConsider the continuous-time system (13. 1)where X=state vector (n-vector) u=control signal (scalar) A= matrix B= matrix The system described by Equation (13. 1) is said to be state controllable atif it ispossible to construct an unconstrained control signal that will transfer an initial state to anyfinal state in a finite time interval. If every state is controllable, then the system issaid to be completely state controllable. We shall now derive the condition for complete state of controllability. Without loss ofgenerality, we can assume that the final state is the origin of the state space and that the initialtime is zero,or.The solution of Equation (13. 1) isApplying the definition of complete state controllability just given, we haveor (13. 2)Andcan be written (13. 3)Substituting Equation (13. 3) into Equation (13. 2) gives (13. 4)Let us putThen Equation (13. 4) becomes (13. 5)If the system os completely state controllable, then, given any initial state X(0), Equation(13. 5) must be satisfied. This requires that the rank of thematrixbe n. From this analysis, we can state the condition for complete state controllability as follows.The system given by Equation (13. 5) is completely state controllable if and only if the vectorsare linearly independent, or thematrixis the rank n.The result just obtained can be extended to the case where the control vector U isr-dimensional. If the system is described byWhere U is an r-vector, then it can be proved that the condition of for complete statecontrollability is that thematrixbe of rank n, or contain n linearly independent column vectors. The matrixis commonly called the controllability matrix. Complete Observability of Continuous-Time System In this section we discuss the observability of linear systems. Consider the unforcedsystem described by the following equations (13. 6) (13. 7)where X=state vector (n-vector) Y=output vector (m-vector) A=matrix C=matrix The system is said to be completely observable if every statecan be determined fromthe observation of Y(t) over a finite time interval,.k The system is, therefore,completely observable if every transition of the state eventually affects every element of theoutput vector. The concept of observability os useful in solving the problem or reconstructingunmeasurable state variable from measurable variables in the minimum possible length of time.In this section we treat only linear, time-invariant systems. Therefore, without loss ofgenerality, we can assume that. The concept of observability is very important because, in practice, the difficultyencountered with state feedback control is that some of the state variables are not accessible fordirect measurement, with the result that it becomes necessary to estimate the unmeasurablestate variables in order to construct the control signals.l Such estimates of state variables arepossible of and only if the system is completely observable. In discussion observability conditions, we consider the unforced system as given byEquation (13. 6) and (13. 7). The reasons for this are as follows, If the system is described bythenAnd Y(t) isSince the matrices A, B, C, and D are known and u(t) is also known,the last terms onthe right-hand side of this last equation are known quantities. Therefore, they may besubtracted from the observed value of Y(t). Hence, for investigating a necessary and s