LPV Model Reduction for a Stable LPV System:

Consider a third-order parameter dependent system:

$$
\begin{array}{l@{}l}
\left[ \begin{array}{c} \dot x_1 (t) \\ \dot x_2 (t) \\ \dot x_{act} \end{array} \right]
 &{}=  \left[ \begin{array}{ccc}
0 & 1 & 0\\
-\omega_n^2 & -2\zeta(\rho)\omega_n & 1\\
0 & 0 & -100
\end{array} \right]
\left[ \begin{array}{c}  x_1 (t) \\ x_2 (t) \\ x_{act} \end{array} \right]
+
\left[ \begin{array}{c}  0 \\ 0 \\ 100 \end{array} \right] u \\
y &{}= x_1
 \end{array} \ \ \ \ \ \ \ (1)$$

where the parameter $\rho$ lies in the interval $[1,5]$, and the coefficients $\zeta(\rho)$ and $\omega_n$ are defined as:

$$\zeta(\rho) = \frac{\sqrt{6-p}}{10} \ \ \ \ \ \ \ (2)$$

$$ \omega_n = 5 \ \ \ \ \ \ \ (3)$$

We note that the system consists of a second order system, with damping coefficient $\zeta(\rho)$ and natural frequency $\omega_n$, that is in series with a first-order actuator at its input. The actuator has a pole at 100 rad/s, which is two orders of magnitude higher than the natural frequency of the second-order system. Hence, if the dynamics at low frequency are the main object of interest, it is possible to remove the actuator state from this 3 state model, with minimal effect in the frequency band where the second-order dynamics take place. Lets do this using LPVTools.

% Define the time-varying rho parameter as a gridded real parameter:
p = pgrid('p',1:5);

% Define the second-order system:
zet = sqrt(6-p)/10;
wn = 5;
G = ss([0 1;-wn^2 -2*zet*wn],[0;1],[1 0],0);

% Define the first order actuator:
act = ss(-100,100,1,0);

% Define the plant model which consists of the the second-order system and
% the actuator in series:
sys = G*act
PSS with 3 States, 1 Outputs, 1 Inputs, Continuous System.   
The PSS consists of the following blocks:                    
  p: Gridded real, 5 points in [1,5], rate bounds [-Inf,Inf].

Lets compare the frequency response of sys and G at the five grid points: $\rho = [1,2,3,4,5]$

freq = linspace(1,1e3,5000);
bode(sys,'b',freq)
hold on
bode(G,'r--',freq)
legend('G*act','G')
hold off

We note the actuator pole kicking in at 100 rad/s, and that its effect on the second order dynamics is negligable. Hence, it should be safe to remove 1 state from the model if we are only interested in the system's dynamics at low frequency.

lpvbalancmr will compute a balanced realization of the LPV system sys and then remove those states that contribute least to its input/output behaviour. sys has 3 states, we will call on lpvbalancmr to remove 1 state and generate a balanced realization with only 2 states.

% The first input to |lpvbalancmr| is the system to be reduced. The second input
% is the desired state order of the output:
[sys_red,info] = lpvbalancmr(sys,2);

sys_red is the 2 state reduced-order model:

sys_red
PSS with 2 States, 1 Outputs, 1 Inputs, Continuous System.   
The PSS consists of the following blocks:                    
  p: Gridded real, 5 points in [1,5], rate bounds [-Inf,Inf].

info stores results for the model reduction. Lets look at the relative size of the Hankel singular values associated with the states in the balanced version of sys:

info.StabSV
ans =
    0.1115
    0.0915
    0.0006

We see that one of the states has a Hankel Singular value that is two orders of magnitude smaller than the smallest Hankel Singular value of the other two. This indicates that it removing this state will have a minimal effect on the accruacy of the resulting reduced order model.

Lets compare the frequency response of the original three state system sys, and the reduced order second-order system sys_red.

freq = linspace(1,1e2,5000);
bode(sys,'b',freq)
hold on;
bode(sys_red,'k:',freq)
legend('sys: Original 3-state model',...
'sys_red: 2-state reducted order model','location','northeast')

We note that the frequency response in of the original three state system and the reduced order system is identical up to approximatly 30 rad/s.

References

  1. G. D. Wood, "Control of parameter-dependent mechanical systems," Ph.D. Dissertation, University of Cambridge, 1995.
  2. G. D. Wood, P. J. Goddard, and K. Glover, "Approximation of linear parameter-varying systems," IEEE Conference on Decision and Control, Vol. 1, pp 406-411, 1996.
  3. R. Widowati, R. Bambang, R. Sagari, S. M. and Nababan, “Model reduction for unstable LPV system based on coprime factorizations and singular perturbation,” 5th Asian Control Conference, Vol. 2, pp. 963-970, Melbourne, Australia, 2004.