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EXAM ON MPC
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Institut fur Automatik D-ITETETH Zurich SS 2012Prof. Dr. M. Morari 22. 03. 2012
MODEL PREDICTIVE CONTROL
Exam
Stud.-Nr. :
Name :
Do not use pencils or red color.
Make sure that your name and student number is
on every sheet you hand in.
Use seperate sheets for the four parts.
Part Points max.1 402 403 254 20
125
1. PartQuestion a) b) c) Total
Max. Points 12 12 16 40Achieved Points
Optimal Control of Linear Systems
a) Consider the following discrete-time system:
xk+1 =
[0 1/2
3/2 2]
A
xk +
[1b
]B
uk
yk =[0 2
] C
xk
(1)
with parameter b R.i) For which b is the system open-loop stable?
ii) For which b is the system controllable?
In the following, let b = 0. The goal is to design a linear state feedbackcontroller uk = Kxk with K =
[k1 k2
]such that from any initial
state x0 the closed-loop system reaches the origin in finite time. Thisis achieved if K is chosen such that all eigenvalues of the closed-loopsystem are zero.
iii) Give a sufficient condition on a general pair A and B for the existenceof such a K. Is it fulfilled for A and B given in (1) with b = 0? Afterhow many steps, at most, does the system arrive at the origin withsuch a controller?
b) We want to design a state observer for system (1).
i) Derive the update equation for the state estimate x such that theerror dynamics are given by
ek+1 = (A LC)ek . (2)
where ek is the estimation error at time step k which is defined as
ek := xk xk . (3)
ii) How many states does the closed-loop system with observer and thecontroller uk = Kxk have in total?
iii) Let L =[1/4 1]T and V (x) := xPx with P = [13/4 0
0 1
].
Show that V is a Lyapunov function for the error dynamics (2). Whatdoes this imply?
Model Predictive Control SS 2012
c) Dynamic Programming. Consider the finite horizon discounted LQR pro-blem
minX,U
N1k=0
k(xTkQxk + u
TkRuk
)such that xk+1 = Axk +Buk ,
(4)
with discount factor (0, 1), Q = QT , Q 0, R = RT and R 0.Assume the following form of the optimal cost-to-go at timestep n, n {0, 1, ..., N} for the discounted problem (4)
Jdi,n (xn) = nxTnP
din xn
where P din = (Pdin )
T and P din 0.i) With the given form of the optimal cost-to-go of the discounted
problem and using the principle of optimality derive the recursionP din+1 P din .
ii) Show that the recursion for P di coincides with the standard Riccatirecursion for P un
P unn = ATP unn+1A ATP unn+1B
(BTP unn+1B + R
)1BTP unn+1A+Q ,
of the undiscounted problem
minX,U
N1k=0
xTkQxk + uTk Ruk
subject to xk+1 = Axk +Buk
with A =A and R = R/.
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Model Predictive Control SS 2012
2. Part
Question a) b) c) TotalMax. Points 12 16 12 40
Achieved Points
Optimization
a) i) Every subspace is a cone
True False
ii) Every affine set is a cone
True False
iii) Every cone is an affine set
True False
iv) The finite intersection of polytopes is always a polytope
True False
v) The finite union of polytopes is always a polytope
True False
vi) Let fi(x) : Rn R, i = 1, . . . , N be a set of N convex functions.Show that the set
S := {x Rn | fi(x) 0, i = 1, . . . , N}
is convex.
b) Consider the following Linear Program
min cTxsubj. to Gx h (5)
where c Rn, G Rmn, h Rm.i) Problem (5) is always convex
True False
ii) Its convexity depends on c
True False
iii) It is always feasible
True False
iv) Its feasibility depends on c
True False
v) Let the matrices in (5) be
c =
[11], G =
1 00 11 00 1
, h =
1111
.Find the optimal solution of (5) and show that it satisfies the KKTconditions.
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Model Predictive Control SS 2012
vi) The optimal solution of v) remains optimal if we change the costvectorto c =
[1 0]TTrue False
vii) The optimal solution of (v) remains optimal if we change the right
hand side of the constraints to h =[1 2 1 1
]TTrue False
c) Consider the following quadratic program with equality constraints
min xTxsubj. to Ax = b
(6)
i) Formulate the Lagrange function corresponding to (6)
ii) Formulate the dual function corresponding to (6)
iii) Formulate the dual optimization problem corresponding to (6) byminimizing (infimizing) over x
iv) Let A =[1 0
]and b = 1 in (6). Solve the primal and dual problem
and show that the duality gap is zero.
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Model Predictive Control SS 2012
3. Part
Question a) b) TotalMax. Points 17 8 25
Achieved Points
Model Predictive Control
Consider the following finite-horizon discrete-time optimal control problemwith Q 0,P 0, R 0, (A,B) controllable:
V (x) = min{u0,...,uN1}
N1k=0
(x>kQxk + u>k Ruk) + x
>NPxN
subject to:
xk+1 = Axk +Buk, x0 = x
Cxk +Duk f, for k {0, . . . . , N 1}
(7)
a) Suppose that N = 3 and that the control inputs uk are modeled as
uk = Kxk + vk,
where K is a constant matrix and the vector vk is a decision variable inthe optimization problem. Define
v :=
v0v1v2
.Find matrices E and S, vectors g and h, and a constant c such that theoptimal control problem (7) can be rewritten as
minv
[v>Sv + h>v + c
]subject to: Ev g.
(8)
b) Assume that problem (7) has no constraints, and that one solves, at eachtime step, the optimization problem
V (x) = minv
[v>Sv + h>v + c
],
with optimal solution
v(x) = (v0(x); v1(x); v
2(x)).
Assume that the matrix K is chosen such that (A+BK) is stable.
Suggest a condition on the matrix P such that the closed-loop system
x(k + 1) = (A+BK)x(k) +Bv0(x)
is stable.
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Model Predictive Control SS 2012
4. Part
Question a) b) TotalMax. Points 14 6 20
Achieved Points
Parametric Linear Program / Hybrid MPC
a) Consider the parametric linear program
Jp (x) = maxz1,z2
z1+xz2
s.t. z1 + z2 = 1 (ppLP)
z1 0z2 0 ,
with parameter x R.i) Sketch the feasible set of (ppLP) and the cost gradients for parameter
values x {0, 1, 2}.ii) Derive the optimizer function z(x) = (z1(x), z
2(x)) and the value
function Jp (x) of (ppLP) for parameter values x [0, 2].Hint: Do it graphically.
iii) The dual program of (ppLP) is given by
Jd (x) = min
s.t. 1 (dpLP) x
Derive the dual optimizer function (x) and the dual value functionJd (x) of (dpLP) for parameter values x [0, 2].
iv) Compare Jp (x) with Jd (x). State the reason why or why not the
value functions coincide.
b) Consider the discrete-time dynamic system
xk+1 = Axk +Buk , k 0 , (SYS)
with state xk Rn and discrete input uk{v, w}, where v, w are vectorsin Rm.
i) Let us represent (SYS) as a mixed logical dynamical (MLD) system.For this, we introduce binary variables (1,k, 2,k) {0, 1}2 for everytime-step k 0 and rewrite (SYS) as
xk+1 = Axk +Bh
[1,k2,k
], k 0
(MLDSYS)
c = d11,k + d22,k , k 0 .
State the input matrix Bh Rn2 and the coefficients (c, d1, d2) R3so that (MLDSYS) is an equivalent description of (SYS).
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Model Predictive Control SS 2012
ii) Suppose now that we want to design a model predictive controller ba-sed on the MLD representation in (MLDSYS), i.e. at every samplinginstant we solve
minx,1,2
1
2xTNPxN+
N1k=0
1
2xTkQxk +
N2k=0
lu(1,k+1, 1,k)
s.t. (MLDSYS), k = 0, . . . , N 1fp(1) 0x0 = x(0) ,
where x(0) Rn is the initial state of the system and x, 1, 2 denotethe sequences of states/binaries over the prediction horizon of lengthN . Design the functions lu, fp such that
changing the discrete input gets penalized, input v is applied to the system at most N/2 times over the
prediction horizon (assuming N is an even number).
Hint: It suffices to restrict lu and fp to the class of affine and qua-dratic functions.
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