By Andrew D. Lewis
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Additional info for A Mathematical Approach to Classical Control
In doing this, we arrive at the following definition. 10 Definition Let (f , h) be a SISO nonlinear system and let (x0 , u0 ) be an equilibrium point for the system. 5) about (x0 , u0 ) is the SISO linear system 32 2 State-space representations (the time-domain) 22/10/2004 (A, b, ct , D) where ∂f1 ∂x2 ∂f2 ∂x2 ··· ··· .. ∂f1 ∂xn ∂f2 ∂xn ∂fn ∂fn ∂x1 ∂f ∂x2 1 ∂u ∂f2 ∂u ··· ∂fn ∂xn ∂f1 ∂x1 ∂f2 ∂x1 A = Df (x0 , u0 ) = . .. b= .. .. . ∂f (x0 , u0 ) = .. ∂u . ∂fn ∂u ∂h ∂x1 ct = Dh(x0 , u0 ) = ∂h D= (x0 , u0 ), ∂u ∂h ∂x2 ∂h ∂xn ··· where all partial derivatives are evaluated at (x0 , u0 ).
3 Bad behaviour due to unstable zero dynamics . . . . . . . . . . 4 A summary of what we have said in this section . . . . . . . . . . 47 The impulse response . . . . . . . . . . . . . . . . . . . 1 The impulse response for causal systems . . . . . . . . . . . . 2 The impulse response for anticausal systems . . . . . . . . . . . 53 Canonical forms for SISO systems . . . . . . . . . . . . . . . . 5 Controller canonical form .
19, for the dynamics to be unstable, even though they are fine for some initial conditions. And these unstable dynamics are not something we can get a handle on with our inputs; this being the case because of the lack of controllability. 3. 27. The problem here is that all the input energy can be soaked by the unstable modes of the zero dynamics, provided the input is of the right type. It is important to note that if we have any of the badness of the type listed above, there ain’t nothing we can do about it.
A Mathematical Approach to Classical Control by Andrew D. Lewis