User:Margav06/sandbox/Click here to continue/Applications of Linear systems/Helicopter Inner Loop LMI

This is a Helicopter Inner Loop LMI. Optimization methods and optimal control have had difficulty gaining traction in the rotorcraft control law community. However, this LMI derived in the referenced paper attempts to address the issues with a LMI for Robust, Optimal Control.

The System
Continuous Time:

\begin{align} \dot x(t)&=\hat{A}x(t)+B_1w(t)+\hat{B_2}u(t),\\ z(t)&=C_1x(t)+D_{12}u(t)\\ \end{align}$$

The Helicopter model is given by knowledge of the stability and control derivatives which populate the elements of the $$ \hat{A}, \hat{B_2} $$ matrices in the dynamic equations above.

The state vector is given by the typical elements of a rigid 6-DOF body model. $$ x = \begin{bmatrix} u,w,q,\theta,v,p,\psi,r\end{bmatrix}^T $$. The input vector is given by $$ u = \begin{bmatrix}\delta_0,\delta_{1s},\delta_{1c},\delta_T\end{bmatrix}^T $$ which pertain to the main rotor collective, longitudinal/lateral cyclic and tail rotor collective blade angles in radians.

The gust disturbance is denoted by $$w(t)$$ and is assumed to be random in nature. The stability and control derivative matrices are modeled with uncertainty as follows:



\begin{align} \hat{A} = A + \Delta A, \hat{B_2} = B_2 + \Delta B\\ \end{align}$$

The $$\Delta $$ terms represent the uncertainties in the helicopter system model.

The Data
The Data required for this LMI are the stability and control derivatives that populate the A and B-matrices of the system above which can be obtained from linearizing non-linear models. It can also be obtained from experimental methods such as step responses and swept sines (System Identification.)

The Control Architecture
A control architecture for the inner loop of the helicopter model mentioned above is designed using a state feedback control law.

$$u = Kx(t) $$

The objective for the inner loop control is to design a full state feedback law such that the closed-loop helicopter system satisfies the following 3 performance specifications.

The Optimization Problem
Objective 1: The closed-loop system is internally stable for any admissible uncertainty.

Objective 2: Poles of the close-loop system lie within the disk $$ D(-q,r)$$ with center $$-q+j_0$$ and radius $$r, q>r>0$$, for any admissible uncertainty.

Objective 3: Given gust disturbance suppression index $$\gamma$$, for any admissible uncertainty, the effect of the gust disturbance to selected flight states and control input is in the given level, i.e.

$$\int_0^{\infty} \! \{x^T(t)Qx(t)+u^T(t)Ru(t)\} \ \mathrm{d}x < \gamma^2\int_0^{\infty} \! w(t)^Tw(t) \, \mathrm{d}t.$$

where $$w(t)\in L_2(0,\infty).$$ $$Q$$ and $$R$$ are weighting matrices with appropriate dimensions and $$Q=Q^T \geq 0, R = R^T>0.$$

It can be shown that the inner loop performance specifications listed in Objectives 1-3 can be met with a state feedback control law if the LMI described in the following section is true.


 * Objective: Objectives listed above
 * Variables: Controller Gains
 * Constraints: Rotorcraft Dynamics and Modeled Actuator Limits

The LMI: H-Inf Inner Loop D-Stabilization Optimization
The paper derives and LMI of the form below and asserts that the if there exists a constant $$\epsilon$$, matrix $$Z$$ with appropriate dimensions and a symmetric positive matrix $$P$$, such that



\begin{align} \\

\begin{bmatrix} \Psi_{11} & B_1 & XC_1^T+Z^TD_12^T & A_{\alpha}X_+B_2Z & XF_1^T+Z^TF_2^T & \epsilon H \\ B_1^T & -\gamma ^2 I & 0 & 0 & 0 & 0 \\ C_1X+D_{12}Z & 0 & -I & 0 & 0 & 0 \\ XA_{\alpha}^T + Z^TB_2^T & 0 & 0 & -rX & XF_1^T+Z^TF_2^T & 0 \\ F_1X+F_2Z & 0 & 0 & F_1X+F_2Z & -\epsilon I & 0 \\ \epsilon H^T & 0 & 0 & 0 & 0 & -\epsilon I \end{bmatrix} < 0 \\

\end{align}$$ where, $$\Psi_{11} = A_{\alpha}X+XA_{\alpha}^T+B_2Z+Z^TB_2^T,A_{\alpha}=A+\alpha I $$

This LMI is shown to satisfy Objectives 1, 2,3, and the control law is given by

\begin{align} u(t) = Kx(t)\\ K = ZX^{-1} \end{align} $$

Conclusion:
The LMI for Helicopter Inner Loop Control design provides an optimization-based approach towards achieving Level 1 Handling Qualities per ADS-33E. This is an interesting way to approach a very difficult problem that has usually been approached through classical control methods and with extensive piloted simulation and flight test.

Implementation
A link to CodeOcean or other online implementation of the LMI

Related LMIs
Links to other closely-related LMIs


 * Optimal_Output_Feedback_Hinf_LMI