Description
1 Dynamic Matrix Control (DMC) uses a specific kind of plant model to make predictions for
a receding-horizon optimal control implementation. Do a little research and write a 1-page
(plus at most 5 references) summary of DMC, focusing on the plant model, the history and
the applications.
2 When forward Euler discretization is applied to a linear state-space system ˙x = Ax, the
following discrete equivalent is obtained:
x
+ = (I + δA)x
where δ is the sampling interval. Find the continuous/discrete eigenvalue mapping as done in
class for backward Euler and comment on any stability preservation properties upon forward
Euler discretization.
3 (mini-project)
A mechanical system (robotics) model without gravity effects is given by
M(q)¨q + C(q, q˙) ˙q = u
where q is the vector of joint coordinates, M(q) is the mass matrix (always symmetric and
positive-definite) and C(q, q˙) is a matrix containing Coriolis and centripetal effects. The
control torque vector is u.
A nonlinear state-space model can be obtained by defining the state vector as z
T = [z
T
1
z
T
2
] =
[q
T
q˙
T
]. With these definitions, the state-space model is
z˙1 = z2
z˙2 = M−1
(z1)(−C(z1, z2)z2 + u)
Symbolic Matlab code has been provided to obtain and evaluate ˙z2, and a Simulink template
to simulate the nonlinear dynamics has also been provided.
1. Note that any state of the form [z
T
1 0] is an equilibrium point with u = 0. Use Matlab’s
diff to linearize the state-space model at q
T = [0 π/4 0 π/4]. Obtain continuous-time
matrices A and B for the linearization.
2. Use nilpotent matrix approach to obtain the ZOH discrete equivalent Γ and Φ sampled
at Ts = 0.01 s.
3. You now have 3 models. Test the accuracy of the linearization and the discretization
by applying the test torque inputs provided in the template to each model. Note that
the linearized system and its discretization use incremental states and inputs, therefore
appropriate shifting must be used.
4. Show plots of each one of the eight states, as obtained by each model.
5. Design a discrete LQR controller to stabilize the discretized system at the equilibrium.
Use Q = I and R = 0.1I as initial tuning. Test the operation of this controller against
the discretized system, the linearized C-T system and finally the nonlinear system.
Shift as required. Simulate the deadbeat regulator applied to the nonlinear system
using some initial condition near the equilibrium condition in use. Informally run some
simulations to see how far the initial conditions can be from the equilibrium without
losing stability.
4 Reproduce the procedure described in class to find the largest invariant ellipsoid contained
in a set of box constraints for the double-integrator plant.
Note: Running code must be submitted for 3 and 4.
