System with Uncertain Parameters

As an example of a closed-loop system with uncertain parameters, consider the two-cart "ACC Benchmark" system [13] consisting of two frictionless carts connected by a spring shown as follows.

ACC Benchmark Problem

The system has the block diagram model shown below, where the individual carts have the respective transfer functions.

$$\begin{array}{l}{G}_1\left(s\right)=\frac{1}{m_1{s}^2}\\ G_2\left(s\right)=\frac{1}{m_2{s}^2}.\end{array}$$

The parameters m₁, m₂, and k are uncertain, equal to one plus or minus 20%:

m1 = 1 ± 0.2 
m2 = 1 ± 0.2 
k = 1 ± 0.2

"ACC Benchmark" Two-Cart System Block Diagram y₁ = P(s) u₁

The upper dashed-line block has transfer function matrix F(s):

$$F\left(s\right)=\left[\begin{array}{c}0\\ G_1\left(s\right)\end{array}\right]\left[\begin{array}{cc}1& -1\end{array}\right]+\left[\begin{array}{c}1\\ -1\end{array}\right]\left[\begin{array}{cc}0& G_2\left(s\right)\end{array}\right].$$

Build the Uncertain Model

To build the uncertain model of the ACC Benchmark two-cart system, first create uncertain real parameters for the masses and spring constant.

m1 = ureal('m1',1,'percent',20);
m2 = ureal('m2',1,'percent',20);
k  = ureal('k',1,'percent',20);

Next, create the transfer functions for each cart.

s = zpk('s');
G1 = ss(1/s^2)/m1;
G2 = ss(1/s^2)/m2;

Use model arithmetic to assemble F(s), and connect F(s) to the spring k.

F = [0;G1]*[1 -1]+[1;-1]*[0,G2];
P = lft(F,k)

Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 4 states.
The model uncertainty consists of the following blocks:
  k: Uncertain real, nominal = 1, variability = [-20,20]%, 1 occurrences
  m1: Uncertain real, nominal = 1, variability = [-20,20]%, 1 occurrences
  m2: Uncertain real, nominal = 1, variability = [-20,20]%, 1 occurrences

Type "P.NominalValue" to see the nominal value and "P.Uncertainty" to interact with the uncertain elements.

The resulting P is a SISO uncertain state-space (uss) object with four states and three uncertain parameters, m1, m2, and k. You can obtain the nominal plant from the NominalValue property.

Pnom = zpk(P.NominalValue)

Pnom =
 
        1
  -------------
  s^2 (s^2 + 2)
 
Continuous-time zero/pole/gain model.

Suppose that the uncertain model P(s) has a negative feedback controller with the following transfer function.

$$\mathit{C}\left(\mathit{s}\right)=\frac{{100\left(\mathit{s}+1\right)}^3}{{\left(0.001\mathit{s}+1\right)}^3}$$

Form the closed-loop transfer function from r to y1 with P(s) and P(s) in the standard configuration.

C = 100*ss((s+1)/(.001*s+1))^3; 
T = feedback(P*C,1); 

View the closed-loop step response on the time interval from t = 0 s to t = 0.1 s. Use usample to generate Monte Carlo random samples of ten combinations of the three uncertain parameters and plot the response. The result gives a sense of the range of responses and the robustness of the system against variations in the uncertain values.

stepplot(usample(T,10),.1);