By Ricardo Zavala Yoe

A paradigmatic viewpoint for modeling and keep an eye on of actual structures has been used due to the fact decades in the past: The input/output technique. even if rather average for our human adventure, this attitude imposes a cause/effect framework to the procedure below research even if such method would possibly not have inevitably such cause/effect constitution. truly, from a common perspective, a method interacts with its setting through alternate of mass and effort which may still indicate using bidirectional arrows in a block diagram instead of utilizing unidirectional ones. one other viewpoint arose while a brand new variable confirmed up in structures and keep watch over thought: the kingdom (Kalman). therefore, the input/state/output strategy used to be born. even supposing the idea that of country is cornerstone, qualitative features of a approach (stability, controllability and observability) need to be outlined by way of a illustration of the procedure, i.e., such features develop into illustration established. by contrast, the rather new Behavioral technique for platforms and keep watch over (Jan C. Willems) bargains at once with the answer of the differential equations which signify the procedure. This set of allowed ideas is often called the habit of the method. therefore, the hot button is the habit and never the illustration. This truth makes this attitude a illustration loose approach.

This e-book studies recognized themes of the Behavioral procedure and provides new theoretic effects with the benefit of together with keep watch over algorithms applied numerically within the machine. furthermore, problems with numerical research also are incorporated. The courses and algorithms are MATLAB based.

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B(ξ) depends only on B, and not on the polynomial matrix R(ξ) we used to define it. If R1 (ξ), R2 (ξ) both represent B minimally then there exists a unimodular U (ξ) such that R2 (ξ) = U (ξ)R1 (ξ). Consequently, if det(R1 (ξ)) and det(R2 (ξ)) are monic then det(R1 (ξ)) = det(R2 (ξ)). Finally, the roots of the characteristic polynomial are called the poles of the autonomous behavior B. 8 Defining inputs and outputs As we have pointed out up to now, the behavioral point of view is a representation free approach and hence is not subjected to any special framework to explain the way a system interacts with its environment.

Of course, there are many behaviors that are wedged in between N and (Pfull )w . Obviously, any manifest controlled behavior (Kfull (C))w must be contained in the manifest plant behavior (Pfull )w . Also, any manifest controlled behavior (Kfull (C))w must contain the hidden behavior N , since if the controller receives no information about what is happening in the plant, N must remain possible in the controlled behavior, independently of the controller we have. Now, it turns out that also the converse holds.

Define R2 (ξ) = U (ξ)R1 (ξ). Denote the behaviors associated with R1 (ξ) and R2 (ξ) by B1 and B2 , respectively. Then: a. B1 ⊆ B2 . b. If, in addition, U (ξ) is unimodular, then B1 = B2 . The converse version is as follows, see [58]. Theorem 4 The full row rank polynomial matrices R1 (ξ), R2 (ξ) ∈ Rg×w [ξ] represent the same behavior B if and only if there exists a unimodular matrix U (ξ) ∈ Rg×g [ξ] such that R1 (ξ) = U (ξ)R2 (ξ). Theorem 5 Let B1 , B2 ∈ Lw be represented by kernel representations R1 dtd w = 0 and R2 dtd w = 0, respectively.

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