Mpc12 Exam

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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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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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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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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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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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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