different control input on linearized and nonlinear system

Question

I have a nonlinear sytem, i need to design the controller for the linearized system and then apply the same controller to the nonlinear system it self to see how it performs, i've done so, but i get totally opposite control input: -88 and + 35

this is the system:

$$\dot{x_1}= x2$$ $$\dot{x_2} = -10x_1+1.8{x_1^2}-0.25x_2-30 +u$$

the equilibrium points without any inputs $$u=0$$ are :$$(-2.16,0) , (7.71,0) $$ (stable and saddle points)

I linearize near the saddle point and get: Matrix A= \begin{bmatrix}0&1 \\ 17.75&-0.25\end{bmatrix}

$B= [0 , 1] '$ $C=[1, 0]$ (so the output is $ y=x_1 $

To design my control i use pole placement:

clc
clear all
 
A=[0 1;17.75 -0.25]
B=[0 1]'
C=[1 0]
D=zeros(1,1);
 
sys = ss(A,B,C,D);

G = tf(sys)

step(G);

PO=10;
ts=5;
zita = abs(log(PO/100))/(sqrt(pi^2 + (log(PO/100))^2)); % scelgo ts e po = sovraelongazione
wn = 4/(zita*ts);
alpha = -zita*wn;
wd = wn*sqrt(1-zita^2);
p1 = alpha + j*wd;
p2 = alpha - j*wd;
p3=10*real(p2)

Aaug = [A zeros(2, 1); -C 0];
Baug = [B;0];
Caug = [C 0];
Daug = D;


K=place(Aaug,Baug,[p1 p2 p3]);

Kp=K(1,1:2);
Ki=K(1,3);

so my control look something like: Simulink scheme of control and my y does arrive to the desired reference input. But the control input is at steady state = -88.

Now if i apply the very same control scheme to the nonlinear system: Nonlinear control scheme where the nonlinear block is just:

 function xdotdot= fcndot(x,xp,u)

    % define your constants
    ma = -10;
    maa= 1.8;
    md = -0.25;
    delta = -30*1;
  



    % nonlinear set of equations
    
    
 xdotdot = ma*x+maa*x*x+md*xp+delta+u

while i'm near the point i will still have Yref= y but my steady state control input is +35 what is strange is that the has opposit sign, while the error, the states and the output have more or less the same shape:

Error linearized system---

Control input linearized system---

States linearized sysem---

Control input on the nonlinear sytem

States nonlinear system

Error nonlinear system

why i have such difference in the input control if i have all the other thing that looks the same? it's due to an error? it is possible?

Answer 1

Despite both controllers are the same, the plants used in both simulations are different. Naturally, the control input $u$ at steady-state will be different as well.

If your intention is to stabilize the nonlinear system at the saddle point, then try this controller:

$$ u = \omega^2 x_{d} - 1.8 x_{1}^2 - K_{p} x_{1} - K_{d} x_{2} + 30 $$

where $x_{d}$ is the desired $x$ value at steady-state, and $\omega$ is the parameter tuned to guarantee the convergence at 5 seconds

$$ x_{d} = \frac{5}{9} (5 + \sqrt{79}) \approx 7.71 $$

$$ \omega = 1.2 $$

with the proportional gain $K_{p}$ and derivative gain $K_{d}$ designed as

$$ K_{p} = \omega^2 - 10 $$

$$ K_{d} = 2 \omega - 0.25 .$$

Note that the control input $u$ at steady-state will be $0$ if $x_{d}$ is one of the unforced equilibrium points.

If your the desired point $x_{d} = 5$, then the control input $u$ at steady-state will be $5 + 30 = 35$, which is the same result that you obtained from your pole placement feedback with integral action.