# Determine whether the system can be stabilized by the control law

I have a controls homework question I am having trouble figuring out where to start. The questions is Determine whether the system can be stabilized by the control law $u = - {G_1}{x_1} - {G_2}{x_2} % MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabg2 % da9iabgkHiTiaadEeadaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSba % aSqaaiaaigdaaeqaaOGaeyOeI0Iaam4ramaaBaaaleaacaaIYaaabe % aakiaadIhadaWgaaWcbaGaaGOmaaqabaaaaa!411F!$.

The system is $A = \left( {\begin{array}{*{20}{c}}0&0&1&0\\0&0&0&1\\0&{ - 4}&{ - 4}&0\\0&{56}&{16}&0\end{array}} \right);B = \left( {\begin{array}{*{20}{c}}0\\0\\1\\{ - 4}\end{array}} \right);C = \left( {\begin{array}{*{20}{c}}1&1&0&0\end{array}} \right) % MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 % da9maabmaabaqbaeqabqabaaaaaeaacaaIWaaabaGaaGimaaqaaiaa % igdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaG % ymaaqaaiaaicdaaeaacqGHsislcaaI0aaabaGaeyOeI0IaaGinaaqa % aiaaicdaaeaacaaIWaaabaGaaGynaiaaiAdaaeaacaaIXaGaaGOnaa % qaaiaaicdaaaaacaGLOaGaayzkaaGaai4oaiaadkeacqGH9aqpdaqa % daqaauaabeqaeeaaaaqaaiaaicdaaeaacaaIWaaabaGaaGymaaqaai % abgkHiTiaaisdaaaaacaGLOaGaayzkaaGaai4oaiaadoeacqGH9aqp % daqadaqaauaabeqabqaaaaqaaiaaigdaaeaacaaIXaaabaGaaGimaa % qaaiaaicdaaaaacaGLOaGaayzkaaaaaa!5787!$

My question is how exactly do I test to see if only feeding back two states yields a stabilizable system? From what I can understand, it is possible to see whether a system is fully controllable to imply that it is stabilizable. But how can one see which states are stabilizable?

Start by computing the closed-loop system. Then compute the characteristic polynomial of the closed-loop system. Finally use the Routh-Hurwitz criterion to see under what conditions of $G_1$ and $G_2$ the characteristic polynomial will be stable. If no such conditions exist the system cannot be stabilized by the proposed control law.

I did the computations of these using Mathematica and got that the system cannot be stabilized. I will summarize the results, and attach the full computations as a screen shot.

The state matrix of the closed-loop system

$$\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -G_1 & -G_2-4 & -4 & 0 \\ 4 G_1 & 4 G_2+56 & 16 & 0 \\ \end{array} \right)$$

The characteristic polynomial $$s^4+4 s^3+(G_1-4 G_2 -56) s^2-160 s-40 G_1$$

The first column of the Routh table $$\left( \begin{array}{c} 4 \\ 4 G_1-16 G_2-64 \\ 16 \left(160 G_2+640\right) \\ 16 \left(-6400 G_2 G_1-25600 G_1\right) \\ \end{array} \right)$$

There exists no conditions under which all of the last three elements are positive.

• Wow thank you! I will need some time to digest this. Commented Feb 21, 2017 at 2:17
• You are welcome. Also, I realized that we need not go through the trouble of doing the complete Routh analysis. The coefficients of the characteristic polynomial are not all positive, whatever the values of $G_1$ and $G_2$. Commented Feb 21, 2017 at 16:29
• I am placing your answer as the best one. Thank you again! In case you are interested, I have another controls problem. I got about 75% of it. Problem is that I do not know how to handle exogenous inputs: electronics.stackexchange.com/questions/288441/… Commented Feb 23, 2017 at 22:29