LMIs in Control/pages/Quadratic DStabilization

Continuous-Time D-Stability Controller

This LMI will let you place poles at a specific location based on system performance like rising time, settling time and percent overshoot, while also ensuring the stability of the system.

The System

Suppose we were given the continuous-time system

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\end{aligned}}

whose stability was not known, and where $A\in \mathbb {R} ^{mxm}$ , $B\in \mathbb {R} ^{mxn}$ , $C\in \mathbb {R} ^{pxm}$ , and $D\in \mathbb {R} ^{qxn}$ for any $t\in \mathbb {R}$ .

Adding uncertainty to the system

{\begin{aligned}{\dot {x}}(t)&=(A+A_{i})x(t)+(B+B_{i})u(t)\\\end{aligned}}

The Data

In order to properly define the acceptable region of the poles in the complex plane, we need the following pieces of data:

matrices $A$ , $B$ , $A_{i}$ , $B_{i}$
rise time ( $t_{r}$ )
settling time ( $t_{s}$ )
percent overshoot ( $M_{p}$ )

Having these pieces of information will now help us in formulating the optimization problem.

The Optimization Problem

Using the data given above, we can now define our optimization problem. In order to do that, we have to first define the acceptable region in the complex plane that the poles can lie on using the following inequality constraints:

Rise Time: $\omega _{n}{\leq }{1.8 \over t_{r}}$

Settling Time: $\sigma {\leq }{-4.6 \over t_{s}}$

Percent Overshoot: $\sigma {\leq }{-ln({M_{p}}) \over {\pi }}|{\omega _{d}}|$

Assume that $z$ is the complex pole location, then:

{\begin{aligned}{\omega _{n}}^{2}=\|z\|^{2}&=z^{*}z\\{\omega _{d}}=Im{z}&={(z-z^{*}) \over 2}\\{\sigma }=Re{z}&={(z+z^{*}) \over 2}\end{aligned}}

This then allows us to modify our inequality constraints as:

Rise Time: $z^{*}z-{1.8^{2} \over {t_{r}}^{2}}{\leq }0$

Settling Time: ${(z+z^{*}) \over 2}+{4.6 \over t_{s}}{\leq }0$

Percent Overshoot: $z-z^{*}+{{\pi } \over ln({M_{p}})}|z+z^{*}|{\leq }0$

which not only allows us to map the relationship between complex pole locations and inequality constraints but it also now allows us to easily formulate our LMIs for this problem.

The LMI: An LMI for Quadratic D-Stabilization

Suppose there exists $X>0$ and $Z$ such that

{\begin{aligned}{\begin{bmatrix}-rP&&AP+BZ\\(AP+BZ)^{T}&&-rP\end{bmatrix}}+{\begin{bmatrix}0&&A_{i}P+B_{i}Z\\(A_{i}P+B_{i}Z)^{T}&&0\end{bmatrix}}<0\\AP+BZ+(AP+BZ)^{T}+A_{i}P+B_{i}Z+(A_{i}P+B_{i}Z)^{T}+2\alpha P&<0,and\\\end{aligned}}

{\begin{aligned}{\begin{bmatrix}AP+BZ+(AP+BZ)^{T}&&c(AP+BZ-(AP+BZ)^{T})\\c((AP+BZ)^{T}-(AP+BZ))&&AP+BZ+(AP+BZ)^{T}\end{bmatrix}}+{\begin{bmatrix}A_{i}P+B_{i}Z+(A_{i}P+B_{i}Z)^{T}&&c(A_{i}P+B_{i}Z-(A_{i}P+B_{i}Z)^{T})\\c((A_{i}P+B_{i}Z)^{T}-(A_{i}P+B_{i}Z))&&A_{i}P+B_{i}Z+(A_{i}P+B_{i}Z)^{T}\end{bmatrix}}<0\\\end{aligned}}

for $i=1,...,k$

Conclusion:

Given the resulting controller $K=ZP^{-1}$ , we can now determine that the pole locations $z\in \mathbb {C}$ of $A(\Delta )+B(\Delta )K$ satisfies the inequality constraints $|x|{\leq }r$ , $Re(x){\leq }-{\alpha }$ and $z+z^{*}{\leq }-c|z-z^{*}|$ for all $\,\Delta \,\in \,C_{0}(\Delta _{1},...,\Delta _{k})$