Since s is a complex variable the cancellation is clearly permissible and we find that the transfer function is GDsE 1 and we have seemingly obtained a contradiction because the system is not equivalent to the system

yDtE uDtE

The problem is easily resolved by using the Kalman representation. In this particular case the system has two subsystems Sro and Sro¯ . The system Sro is a static system with transfer function GDsE 1 and the subsystem Sro¯ which is observable but non reachable has the dynamics.

dxdt x

Notice that cancellations typically appear when using Laplace transforms because of the assumption that all initial values are zero. The consequences are particularly serious when factors like s − 1 are cancelled because they correspond to exponentially growing signals. In the early development of control cancellations were avoided by ad hoc rules forbidding cancellation of factors with zeros in the right half plane. Kalman's decomposition gives a very clear picture of what happens when poles and zeros are cancelled.

3.8 Summary

This chapter has summarized some properties of dynamical systems that are useful for control. Both input-output descriptions and state descriptions are given. Much of the terminology that is useful for control has also been introduced.

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