(Computer Control)8. MPC with Constraints

Model Predictive Control

Model Predictive Control with Constraints

From the last note, we have the augmented state space model

$x(k+1)=Ax(k)+BDelta u(k) \ y(k)=Cx(k) \ Y = Fx(k)+PhiDelta U\ J=frac{1}{2}Delta U^T(Phi^TPhi+ar R)Delta U-Delta U^T Phi^T(R_s-Fx(k))$

If we denote

$H=Phi^TPhi+ar R \ f=-Phi^T(R_s-Fx(k))$

We have

$ag{} J=frac{1}{2}Delta U^THDelta U+Delta U^T f$

Now we will discuss different kinds of constraints.

Constraints on the control variable incremental variation

$ag{} Delta U^{min}leDelta U leDelta U^{max}$

We will have

$Delta Ule Delta U^{max}\ -Delta Ule -Delta U^{min}$

Equally, we have

$ag{} egin{bmatrix} -I \ I end{bmatrix} Delta U le egin{bmatrix} -Delta U^{min} \ Delta U^{max} end{bmatrix}$

2. Constraints on the amplitude of control variable

$overbrace{egin{bmatrix} u(k)\ u(k+1)\ vdots\ u(k+N_p-1) end{bmatrix}}^{U}= overbrace{egin{bmatrix} 1\ 1\ vdots\ 1 end{bmatrix}}^{C_1}u(k-1)+ overbrace{egin{bmatrix} 1 & 0 & dotsm & 0\ 1 & 1 & dotsm & 0\ vdots&vdots&ddots&vdots\ 1&1&dotsm&1 end{bmatrix}}^{C_2} overbrace{egin{bmatrix} Delta u(k)\ Delta u(k+1)\ vdots\ Delta u(k+N_p-1) end{bmatrix}}^{Delta U}$

Equally, we have

$ag{}U=C_1u(k-1)+C_2Delta U$

We can get the following result

$ag{}egin{bmatrix} -C_2 \ C_2 end{bmatrix} Delta U le egin{bmatrix} -U^{min}+C_1u(k-1) \ U^{max}-C_1u(k-1) end{bmatrix}$

3. Output Constraints

$ag{}Y^{min}le Y le Y^{max}$

We have the conclusion form the last note

We can derive the same result as the previous procedures

$ag{}egin{bmatrix} -Phi \ Phi end{bmatrix} Delta U le egin{bmatrix} -Y^{min}+Fx(k) \ Y^{max}-Fx(k) end{bmatrix}$

4. Conclusion

$ag{}egin{bmatrix} -I \ I \ -C_2 \ C_2 \ -Phi \ Phi end{bmatrix} Delta U le egin{bmatrix} -Delta U^{min} \ Delta U^{max} \ -U^{min}+C_1u(k-1) \ U^{max}-C_1u(k-1) \ -Y^{min}+Fx(k) \ Y^{max}-Fx(k) end{bmatrix}$