Description

Problem 1: Consider the following unstable system

(1) Design an H∞ dynamic controller to minimize the H∞ norm of the closed-loop system from w to y and at the same time place all closed-loop poles to the left of a vertical line with real coordinate at −0.2 in the complex plane.
(2) Verify that the closed-loop system meets the designed H∞ norm and pole location constraints.
Problem 2: Consider a mechanical structure modelled as a three mass-spring interconnection where u is the force (control) input. Assuming unit masses, the state-space representation of the system is x˙ = Ax + Bu where
 0 0 0 1 0 0 0
 0 0 0 0 1 0 0
A = −0k1 k01 00 00 00 10, B = 10.

   
 k1 −k1− k2 k2 0 0 0 0
0 k2 −k2 0 0 0 0
The stiffness values k1 and k2 are variable with the nominal values of k1 = k2 = 1, and they vary in the interval [1 − α,1 + α] where the allowable perturbation is α = 0.1.
(1) Confirm that the open-loop uncertain system is not quadratically stable. Notice that the uncertain system can be considered as an affine system since matrix A can be written as A = A0 + k1A1 + k2A2 (Use ”quadstab” in MATLAB).
(2) Design an LQR stabilizing controller for the nominal system using the following MATLAB command:
>> K = −lqr(A,B,eye(6),1) (LQR control with unit weights)
Consider the state-feedback control law u = Kx and compute the closed-loop system. Then, determine if the uncertain closed-loop system is quadratically stable or not when the stiffness vary in the interval as above.
1
(3) Determine the maximum region of quadratic stability of the closed-loop uncertain system, that is, find the maximum α = αmax such that the above uncertain system is quadratically stable for all stiffness perturbations in the interval [1−αmax,1+αmax]. Confirm that the closed-loop system is stable for all upper and lower bounds of the stiffness values ki = 1 − αmax and ki = 1 + αmax (i = 1,2), i.e., confirm stability for all four corner closed-loop systems.
Problem 3: Consider the following system that corresponds to the rigid body motion of a satellite:

Notice that in the above formulation, the system output vector y includes the angular velocity and the control input u. The disturbance input w(t) consists of a plant disturbance and a measurement disturbance signal (and so, it is a vector).
(1) Simulate the response of the open-loop system (i.e., with u = 0) to a pulse disturbance w of amplitude
0.1 and duration of 1 sec (for both the plant and the measurement components).
(2) Use ”hinflmi” in MATLAB to design a dynamic controller that minimizes the energy-to-energy gain of the closed-loop system.
(3) Compute the closed-loop system and examine/validate the closed-loop system stability and performance.
(4) Simulate the closed-loop system output y to a pulse disturbance w of amplitude 0.1 and duration of 1 sec (same input as part (1)).
NOTE: Please include your MATLAB (or Python) files and outputs.
2