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

Frictionless carts with masses m1 and m2 and positions x1 and x2 connected by a spring with constant k. The system input u1 is the control force applied to the first cart, and the system output y1 is the position measurement x2 of the second cart.

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

$\begin{array}{l} G_{1} (s) = \frac{1}{m_{1} s^{2}} \\ G_{2} (s) = \frac{1}{m_{2} s^{2}} . \end{array}$

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

Block diagram showing the interconnections of the cart transfer functions G1 and G2 with the spring k. The diagram expresses the system transfer function P(s) as the LFT interconnection of a block F(s) having inputs {u1,u2} and outputs {y1,y2} with the spring block having input y2 and output u2.

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

$F (s) = [\begin{matrix} 0 \\ G_{1} (s) \end{matrix}] [\begin{matrix} 1 & - 1 \end{matrix}] + [\begin{matrix} 1 \\ - 1 \end{matrix}] [\begin{matrix} 0 & G_{2} (s) \end{matrix}] .$

Build the Uncertain Model

Open Live Script

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.

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

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);

MATLAB figure

System with Uncertain Parameters

Build the Uncertain Model

See Also

Related Topics