LPV Model Reduction for a Unstable LPV System:
Contents
LPV Model Reduction for a Unstable LPV System
Consider a third-order parameter dependent system:
where the parameter lies in the interval
, and the coefficients
and
are defined as:
We note that the system consists of a second order system, with damping coefficient and natural frequency
, that is in series with a first-order actuator at its input. The second order system is unstable for
and marginally stable for
. 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 unstable 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 unstable 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 = (2-p)/30; 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:
freq = linspace(1,1e3,10000); 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.
lpvncfmr will compute a contractive coprime factorization of the LPV system sys, which removes those states that contribute least to its input/output behaviour. sys has 3 states, we will call on lpvncfmr to % remove 1 state and generate a realization with only 2 states. Note that the function lpvbalancmr can not be used in this case, because the LPV system is unstable and it requires a stable LPV system. lpvncfmr can handle both stable and unstable LPV systems.
% The first input to |lpvncfmr| is the system to be reduced. The second % input is the desired state order of the output: sys_red = lpvncfmr(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].
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: 3-state model',... 'sys_red: 2-state 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
- G. D. Wood, "Control of parameter-dependent mechanical systems," Ph.D. Dissertation, University of Cambridge, 1995.
- 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.
- 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.