LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Control of Rational Systems using Linear-fractional Representations and Linear Matrix Inequalities*

Introducation
An optimization-based methodology for the multiobjective control of a large class of nonlinear systems is performed.

The System
Consider a nonlinear, continuous-time system $$\dot{y} = A(x) + B_u(x)u,$$ $$y = C_y(x) +D_{yu}(x)u,$$ where $$x \in \mathbb{R}^{n}$$ is the state vector, $$u \in \mathbb{R}^{n_u}$$ is the input and $$y \in \mathbb{R}^{n_y}$$ is the output.

The Data
$$x \in \mathbb{R}^{n}$$ is the state vector, $$u \in \mathbb{R}^{n_u}$$ is the input and $$y \in \mathbb{R}^{n_y}$$ is the output. $$A, B_u, C_y, D_{yu}$$ are multivariable functions of x. $$A(0) = 0$$ (that is, 0 is an equilibrium point of the unforced system associated with the system). $$C_y(0) = 0$$ and $$B_u, D_{yu}$$ have no singularities at the origin.

Proof
For a given scalar \sigma > 0, we associate with the Linear differential inclusion, $$\dot{x} = Ax + B_uu + B_pp,$$ $$q = C_qx + D_{qu}u + D_{qp}p,$$ $$y = C_yx + D_{yu}u + D_{yp}p,$$ $$p = \bigtriangleup(t)q, \| \bigtriangleup(t) \| \leq \sigma^-1, \bigtriangleup(t) \in D(r), t \geq 0.$$
 * $$\begin{bmatrix} A(x) & B_u(x) \\ C_y(x) & D_{yu}(x) \end{bmatrix}$$ = $$\begin{bmatrix} A & B_u \\ C_y & D_{yu} \end{bmatrix}$$ +  $$\begin{bmatrix} B_p \\ D_{yp} \end{bmatrix}$$$$\bigtriangleup(x)$$$$\begin{bmatrix} C_q & D_{qu} \end{bmatrix}$$

The LMI: Control of Rational Systems using Linear-fractional Representations and Linear Matrix Inequalities
For given \sigma > 0, the LDI system is quadratically stable if there exists P, S, and G such that the LMIs $$P>0, S>0, G = -G^T, S, G \in B(r),$$

$$\begin{bmatrix} A^TP + PA + C_{q}^TSC_q & PB_p + C{q}^TG + C_{q}^TSD_qp \\ B_{p}^TP - GC_q + D_{qp}^TSC_q & D_{qp}^TSD_{qp}G - \sigma^2S + D_{qp}^TG - GD_qp \end{bmatrix} < 0 $$

hold. Then, for every $$\bigtriangleup \in D(r)$$ such that $$\| \bigtriangleup(t) \| \leq \sigma^-1 ,$$ $$det(I - D_{pq}\bigtriangleup)^{-1} \neq 0, $$

$$\begin{bmatrix} y_{max}^2I & C_y \\ C_y^T & P \end{bmatrix} \geq 0.$$

$$\begin{bmatrix} AQ + QA^T + B_pTB_p^T + B_uY + T^TB_u^T & QC_q^T + Y^TD_{qu}^T + B_pTD_{qp}^T + B_pH \\ C_qQ + D_{qu}Y + D_{qp}TB_p^T - HB_p^T & D_{qp}TD_{qp}^T - \sigma^2T + D_{qp}H - HD_{qp}^T \end{bmatrix}$$ < 0

Conclusion
The above LMIs provide a unified setting, as well as an efficient computational procedure, for answering (possibly conservatively) several control problems pertaining to a quite generic class of nonlinear systems. This method makes an explicit and systematic connection (via LFRs and LMIs) between robust control methods and nonlinear systems.