QP_MPC2QP – Conversion of MPC problem to quadratic programming

Block SymbolLicensing group: ADVANCED

Function Description
Quadratic Programming (QP) is a standard technique which suites very well to solve model based predictive control (MPC) problems [9]. Quadratic Programming is an optimization technique that minimizes the sum of quadratic form and linear form.

The QP_MPC2QP block converts a linear MPC problem with quadratic optimization criterion to a quadratic programming problem. The block is compatible with the block QP_UPDATE and the QP solver QP_OASES.

MPC problem formulation

The MPC problem consists of a discrete linear time invariant state space model

\begin{align} \begin{aligned} x_{k + 1} & = A x_{k} + B_{u} u_{k} + B_{v} v_{k}, \\ y_{k} & = C x_{k}, \end{aligned} & (15.1) \end{align}

where $x \in ℝ^{n}$ is the state vector, $u \in ℝ^{m_{u}}$ is the input vector, $v \in ℝ^{m_{v}}$ is the disturbance vector and $y \in ℝ^{p}$ is the output vector. Matrices $A \in ℝ^{n \times n}$ , $B_{u} \in ℝ^{n \times m_{u}}$ , $B_{v} \in ℝ^{n \times m_{v}}$ and $C \in ℝ^{p \times n}$ are referenced by inputs uA, uBu, uBv and uC. The model based predictive control problem is formulated as an optimization problem – minimization of the quadratic optimality criterion (cost function) in the form

J = \sum_{k = 1}^{n_{p}} \{{\hat{x}}_{k}^{T} Q {\hat{x}}_{k} + {\hat{x}}_{k}^{T} W + û_{k - 1}^{T} R û_{k - 1}\},

(15.2)

where symmetric and positive (semi-)definite matrices $Q \in ℝ^{n \times n}$ and $R \in ℝ^{m_{u} \times m_{u}}$ and the vector $W \in ℝ^{n}$ are referenced by inputs uQ, uR and uW, and $n_{p}$ is the prediction horizon (input np).

Additional constraints on the state $x$ and the output $y$ may be required for the minimization process:

\begin{align} x_{\min} & \leq x_{k} \leq x_{\max} & (15.3) \\ y_{\min} & \leq y_{k} \leq y_{\max} & (15.4) \end{align}

For detailed derivation, see the PDF version of this manual.

Inputs

np	Prediction horizon $↓$ 1 $↑$ 1000000	Long (I32)
nc	Control horizon $↓$ 1 $↑$ 1000000	Long (I32)
uA	Input reference to system matrix A	Reference
uBu	Input reference to input matrix Bu of control vector u	Reference
uBv	Input reference to input matrix Bv of disturbance vector v	Reference
uC	Input reference to output matrix C	Reference
uQ	Input reference to symmetric matrix Q in cost function	Reference
uW	Input reference to vector W in cost function	Reference
uR	Input reference to symmetric matrix R in cost function	Reference
ul	Input reference to integer index vector l	Reference
uH	Input reference to Hessian matrix H	Reference
uGx	Input reference to part of gradient vector G corresponding to state vector x	Reference
uGv	Input reference to part of gradient vector G corresponding to disturbance vector v	Reference
uGw	Intput reference to part of gradient vector G corresponding to vector W	Reference
uSuL	Input reference to work matrix Su*L	Reference
uSv	Input reference to work matrix Sv	Reference
uT	Input reference to work matrix T	Reference
uScuL	Input reference to work matrix Scu*L	Reference
uScv	Input reference to work matrix Scv	Reference
uTc	Input reference to work matrix Tc	Reference
uWORK	Input reference to matrix WORK	Reference
HLD	Hold	Bool

Outputs

yA	Output reference to system matrix A	Reference
yBu	Output reference to input matrix Bu of control vector u	Reference
yBv	Output reference to input matrix Bv of disturbance vector v	Reference
yC	Output reference to output matrix C	Reference
yQ	Output reference to symmetric matrix Q in cost function	Reference
yW	Output reference to vector W in cost function	Reference
yR	Output reference to symmetric matrix R in cost function	Reference
yl	Output reference to integer index vector l	Reference
yH	Output reference to Hessian matrix H	Reference
yGx	Output reference to part of gradient vector G corresponding to state vector x	Reference
yGv	Output reference to part of gradient vector G corresponding to disturbance vector v	Reference
yGw	Output reference to part of gradient vector G corresponding to vector W	Reference
ySuL	Output reference to work matrix Su*L	Reference
ySv	Output reference to work matrix Sv	Reference
yT	Output reference to work matrix T	Reference
yScuL	Output reference to work matrix Scu*L	Reference
yScv	Output reference to work matrix Scv	Reference
yTc	Output reference to work matrix Tc	Reference
yWORK	Output reference to matrix WORK	Reference
E	Error indicator	Bool

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