EECE Adaptive Control
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1 EECE Adaptive Control Iterative Control Guy Dumont Department of Electrical and Computer Engineering University of British Columbia January 2010 Guy Dumont (UBC EECE) EECE Iterative Control January / 29
2 Contents 1 Introduction 2 Youla-Kucera Parametrization Open-Loop Inversion Revisited Affine Parametrization: The Stable Case All Stabilizing Controllers 3 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
3 Introduction Guy Dumont (UBC EECE) EECE Iterative Control January / 29
4 Outline Youla-Kucera Parametrization Novel way of expressing controller transfer function Provides new insight into control design The key feature of the new parameterization is that it renders the closed loop sensitivity functions linear or (more correctly, affine) in a design variable We call it the affine parameterization The so-called Youla-Kucera parametrization now plays a central role in control, identification and adaptive control Guy Dumont (UBC EECE) EECE Iterative Control January / 29
5 Youla-Kucera Parametrization Open-Loop Inversion Revisited Open-Loop Inversion Revisited Recall that control implicitly and explicitly depends on plant model inversion. This is best seen in the case of open loop control. In open loop control the input, U(s), is generated from the reference signal R(s), by a transfer function Q(s), i.e. U(s) = Q(s)R(s). This leads to an input-output transfer function of the following form: T 0 (s) = P(s)Q(s) Guy Dumont (UBC EECE) EECE Iterative Control January / 29
6 Youla-Kucera Parametrization Open-Loop Inversion Revisited Open-Loop Inversion Revisited This simple formula highlights the fundamental importance of inversion, as T 0 (jω) will be 1 only at those frequencies where Q(jω) inverts the model. Note that this is consistent with the prototype solution to the control problem described earlier. A key point is that T 0 (s) = P(s)Q(s) is affine in Q(s). On the other hand, with a conventional feedback controller, C(s), the closed loop transfer function has the form T 0 (s) = P(s)C(s) 1 + P(s)C(s) The above expression is nonlinear in C(s). Guy Dumont (UBC EECE) EECE Iterative Control January / 29
7 Youla-Kucera Parametrization Open-Loop Inversion Revisited Comparing the two previous equations, we see that the former affine relationship holds if we simply parameterize C(s) in the following fashion: Q(s) = C(s) 1 + C(s)P(s) We can then design in terms of Q(s) and then obtain C(s) from Q(s) and P(s) This is the essence of the idea presented here. Guy Dumont (UBC EECE) EECE Iterative Control January / 29
8 Youla-Kucera Parametrization All Stabilizing Controllers Affine Parameterization. The Stable Case We can invert the relationship given on the previous slide to express C(s) in terms of Q(s) and P(s): C(s) = Q(s) 1 Q(s)P(s) We will then work with Q(s) as the design variable rather than the original C(s). Note that the relationship between C(s) and Q(s) is one-to-one and thus there is no loss of generality in working with Q(s). Guy Dumont (UBC EECE) EECE Iterative Control January / 29
9 Stability Youla-Kucera Parametrization All Stabilizing Controllers Actually a very hard question is the following: Given a stable transfer function P(s), describe all controllers, C(s) that stabilize this nominal plant. However, it turns out that, in the Q(s) form, this question has a very simple answer, namely all that is required is that Q(s) be stable. This result is formalized in the lemma stated on the next slide. Guy Dumont (UBC EECE) EECE Iterative Control January / 29
10 Youla-Kucera Parametrization All Stabilizing Controllers All Stabilizing Controllers Lemma (Lemma 15.1: Affine parameterization for stable systems) Consider a plant having a stable nominal model G 0 (s) controlled in a one d.o.f. feedback architecture with a proper controller. Then the nominal loop is internally stable if, and only if, Q(s) is any stable proper transfer function when the controller transfer function C(s) is parameterized as Q(s) C(s) = 1 Q(s)P(s) Guy Dumont (UBC EECE) EECE Iterative Control January / 29
11 Youla-Kucera Parametrization All Stabilizing Controllers All Stabilizing Controllers Proof We note that the four sensitivity functions can be written as T 0 (s) = Q(s)P(s) S 0 (s) = 1 Q(s)P(s) S i0 (s) = [1 Q(s)P(s)]P(s) S u0 (s) = Q(s) We are for the moment only considering the case when P(s) is stable. Then, we see that all of the above transfer functions are stable if, and only if, Q(s) is stable. Guy Dumont (UBC EECE) EECE Iterative Control January / 29
12 Youla-Kucera Parametrization All Stabilizing Controllers Youla s parameterization of all stabilizing controllers for stable plants This particular form of the controller, i.e. C(s) = can be drawn schematically below. Q(s) 1 Q(s)P(s) Guy Dumont (UBC EECE) EECE Iterative Control January / 29
13 Youla-Kucera Parametrization All Stabilizing Controllers Conversely let P 0 (s) = N(s)D 1 (s) be a nominal plant stabilized by the controller C(s) = X(x)Y 1 (s) Then all the plants stabilized by C(s) are given by P(s) = N(s) + Y(s)S(s) D(s) X(s)S(s) where S(s) is a stable proper rational transfer function Guy Dumont (UBC EECE) EECE Iterative Control January / 29
14 Youla-Kucera Parametrization Closed-Loop Identification All Stabilizing Controllers Guy Dumont (UBC EECE) EECE Iterative Control January / 29
15 Youla-Kucera Parametrization Closed-Loop Identification All Stabilizing Controllers Guy Dumont (UBC EECE) EECE Iterative Control January / 29
16 The Windsurfer s Approach The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
17 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
18 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
19 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
20 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
21 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
22 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
23 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
24 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
25 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
26 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
27 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
28 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
29 The Windsurfer s Approach Guy Dumont (UBC EECE) EECE Iterative Control January / 29
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