Derive a predictive control law that is based on the following concept. A single control move, Δu(k), is calculated so that the J-step-ahead prediction is equal to the set point, that is, ̂y (k + J) = ysp where integer J is a tuning parameter. This sampling instant, k + J,is referred to as a coinci

Question

Derive a predictive control law that is based on the following concept. A single control move, [latex]\Delta u(k)[/latex], is calculated so that the J-step-ahead prediction is equal to the set point, that is, [latex]\hat{y}(k+J)=y_{sp}[/latex] where integer [latex]J[/latex] is a tuning parameter. This sampling instant, [latex]k + J[/latex], is referred to as a coincidence point. Assume that u is held constant after the single control move, so that [latex]\Delta u(k + i) = 0 for i > 0[/latex].

Accepted Answer

In the proposed predictive control strategy, only a single prediction for J steps ahead is considered. Thus, we let [latex]j = J[/latex] in Eq. 20-20. Similarly, because we are only interested in calculating the current control move, [latex]\Delta u(k)[/latex], the future control moves in Eq. 20-20 are set equal to zero: [latex]\Delta u(k + J − i) = 0 for i = 1, 2 ,…, J − 1[/latex]. Thus, Eq. 20-20 reduces to

[latex]\hat{y}(k+j) =\sum\limits_{i=1}^{j}{S_{i} \Delta u(k+j-i)+\hat{y} ^{\circ }(k+j)}[/latex] (20-20)

[latex]\hat{y}(k+J)=S_{J}\Delta u(k)+\hat{y} ^{\circ } (k+J)[/latex] (20-21)

Setting [latex]\hat{y}(k+J)=y_{sp}[/latex] and rearranging gives the desired predictive controller:

[latex] \Delta u(k)=\frac{y_{sp} -\hat{y} ^{\circ } (k+J)}{S_{J} }[/latex] (20-22)

The predicted unforced response [latex]\hat{y} ^{\circ } (k+J)[/latex] can be calculated from Eq. 20-19 with [latex] j = J[/latex].

[latex]\hat{y} ^{\circ } (k+j)≜\sum\limits_{i=j+1}^{N-1}{S_{i}\Delta u (k+j-i)}+S_{N} u(k+j-N)[/latex] (20-19)

The control law in Eq. 20-22 is based on a single prediction that is made for J steps in the future. Note that the control law can be interpreted as the inverse of the predictive model in Eq. 20-21.

Question 20.2: Derive a predictive control law that is based on the followi...

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