Model Predictive Control (MPC)

MPC

Linear MPC requires solving a Quadratic Program (QP)

$\underset{z}{min}\frac{1}{2}{z}^{\prime }Hz+x(t{)}^{\prime }{F}^{\prime }z+\frac{1}{2}{x}^{\prime }(t)Yx(t)$

s.t. $Gz\le W+Sx$

where $z=\left[\begin{array}{c}{u}_0{u}_1⋮u_{N-1}\end{array}\right]$

These define a polyhedron where the solution should lie.

QP solver

fast gradient method for dual QP

• apply fast gradient method to dual QP

  ​ $\underset{z}{min}\frac{1}{2}{z}^{\prime }Hz+x^{\prime }{F}^{\prime }z$

  ​ s.t. $Gz\le W+Sx$

  Example:

  Model:

  ​ $\begin{array}{rlrlr}{w}_k& =y_k+{\beta }_k(y_k-y_{k-1})z_k& =-K{w}_k-Jx{s}_k& =\frac{1}{L}G{z}_k-\frac{1}{L}(Sx+W)y_{k+1}& =max{y}_k+s_k,0\end{array}$

  ​ Where $y_{-1}=y_0=0$, $K=H^{-1}{G}^{\prime }$ , and $J=H^{-1}{F}^{\prime }$, $\beta =\{\begin{array}{l}0,k=0\frac{k_1}{k+2},k>0\end{array}$

  Termination criterion:

  1. primal feasibility

     $s_k^i\le \frac{1}{L}{ϵ}_G,\mathrm{\forall }i=1,\cdots ,m$

  2. primal oprimality

     $f(z_k)-f^{\ast }\le f(z_k)-\varphi (w_k)=-w_k^{\prime }{s}_{k}L\le {ϵ}_V$

     and $-w_k^{\prime }{s}_k\le \frac{1}{L}{ϵ}_V$

solve MPC:

Step 1: get a linear discrete-time model

system linearization

Step 2: design the MPC controller

LPV

LinearParameter-Varying(LPV) model

​ $$\begin{array}{rlr}{x}_{k+1}& =A(p(t))x_k+B(p(t))u_k+B_v(p(t))v_k{y}_{k+1}& =C(p(t))x_k+D_v(p(t))v_k\end{array}$$

​ depends on $p(t)$

the quadratic perfirmance index can also be LPV, and the resulting optimization problem is still a QP:

​ $$\begin{array}{r}\underset{z}{min}\frac{1}{2}{z}^{\prime }H(p(t))z+{\left[\begin{array}{c}x(t)r(t)u(t-1)\end{array}\right]}^{\prime }F(p(t){)}^{\prime }{z}^{\prime }\text{s.t.}G(p(t))z\le W(p(t))+S(p(t))\left[\begin{array}{c}x(t)r(t)u(t-1)\end{array}\right]\end{array}$$

The QP matrices must be constructed online, contrarily to the LTI case.

An LPV model can obtained by linearizing nonlinear model

​ $\{\begin{array}{l}{\dot{x}}_x(t)=f(x_c(t),u_c(t),p_c(t))y_c(t)=g(x_c(t),p_c(t))\end{array}$

​ where $p_c$ = a vector of exogeneous signals

Linearize:

​ $$\dot{x}c(t) \simeq \underbrace{ \frac{\partial f}{\partial x}|{\bar{x}_c,\bar{u}_c,\bar{p}c}}{A_c} (x_c(t)-\bar{x}c) + \underbrace{ \frac{\partial f}{\partial u}|{\bar{x}_c,\bar{u}_c,\bar{p}c}}{B_c} (u_c(t)-\bar{u}c)+ \underbrace{ \left[ \frac{\partial f}{\partial p}|{\bar{x}_c,\bar{u}_c,\bar{p}_c} f(\bar{x}_c,\bar{u}c,\bar{p}c) \right]}{B{vc}} $$\left[\begin{array}{c}[p_c(t)-{\overline{p}}_{c}1]\end{array}\right]$$ $$

• convert $\left[\begin{array}{c}{A}_c,[B_c{B}_{vc}]\end{array}\right]$ to discrete-time and get prediction model $(A,[]B{B}_{vc})$
• same thing for the output equation, to get matrices $C$ and $D_v$

LTV

Linear time-varying (LTV) model

• $\begin{array}{rlr}{x}_{k+1}& =A_k(t)x_k+B_k(t)u_k{y}_k& =C_k(t)x_k\end{array}$

• at each time the model can also change over the prediction horizon $k$

• the optimization problem is still a QP

  ​ $$\begin{array}{r}\underset{z}{min}\frac{1}{2}{z}^{\prime }H(t)z+{\left[\begin{array}{c}x(t)r(t)u(t-1)\end{array}\right]}^{\prime }F(t{)}^{\prime }{z}^{\prime }\text{s.t.}G(t)z\le W(t)+S(t)\left[\begin{array}{c}x(t)r(t)u(t-1)\end{array}\right]\end{array}$$

• As for LPV-MPC, the QP matrices must be constructed online.

• LTV/LPV model can be obtained by linearing nonlinear models
• nominal trajectories:
  • $U={\overline{u}}_c(t),{\overline{u}}_c(t+T_s),\cdots ,{\overline{u}}_c(t+(N-1)T_s)$
    • $U$=shifted previous optimal sequence, or reference trajectories.
  • $P={\overline{p}}_c(t),{\overline{p}}_c(t+T_s),\cdots ,{\overline{p}}_c(t+(N-1)T_s)$
    • no preview: ${\overline{p}}_c(t+k)\equiv {\overline{p}}_c(t)$