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Preliminaries for Model Predictive Control course

James B. Rawlings

Department of Chemical and Biological EngineeringUniversity of WisconsinMadison

Graduate School inSystems, Optimization, Control and Networks (SOCN)

K.U. LeuvenLeuven, Belgium

August 29September5, 2013

SOCN 2013 MPC short course 1 / 45

Outline

1 Introduction

2 Stability, equilibria, invariant sets

3 Lyapunov function theory

4 Disturbances and robust stability

5 Basic MPC problem and nominal stability


Predictive control

001100110011

00110011

00110011001100110011001100110011 0011 0011

001100110011001100110011

001100110011001100110011 001100110011 001100110011 0011 00110011 00110011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 001100110011

0 00 01 11 1 0 00 00 01 11 11 10011 0011 000111 MeasurementMH Estimate MPC controlForecast

t time

Reconcile the past Forecast the future

sensorsy

actuatorsu

minu(t)

T0|ysp g(x, u)|

2Q

+ |usp u|2R

dt

x= f(x, u)

x(0) =x0 (given)

y =g(x, u)SOCN 2013 MPC short course 3 / 45

State estimation

001100110011

00110011

00110011001100110011001100110011 0011 0011

001100110011001100110011

001100110011001100110011 001100110011 001100110011 0011 00110011 00110011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 001100110011

0 00 01 11 1 0 00 00 01 11 11 10011 0011 000111 MeasurementMH Estimate MPC controlForecast

t time

Reconcile the past Forecast the future

sensorsy

actuatorsu

minx0,w(t)

0T

|y g(x, u)|2R+ |x f(x, u)|2Qdt

x= f(x, u) +w (process noise)

y= g(x, u) +v (measurement noise)



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System model1

We consider systems of the form

x+ =f(x, u)

where the state x lies in Rn and the control (input) u lies in Rm;

In this formulationx and udenote, respectively, the current state and

control, andx+ the successor state.We assume in the sequel that the function f : Rn Rm Rn iscontinuous.

Let

(k; x, u)

denote the solution ofx+ =f(x, u) at time k if the initial state isx(0) =xand the control sequence is u = {u(0), u(1), u(2), . . .};

The solution exists and is unique.1Most of this preliminary material is taken from Rawlings and Mayne (2009,

Appendix B). Downloadable from www.che.wisc.edu/~jbraw/mpc.SOCN 2013 MPC short course 5 / 45

Existence of solutions to model

If a state-feedback control law u= (x) has been chosen, the

closed-loop system is described by x+ =f(x, (x)).

Let (k; x, ()) denote the solution of this difference equation attimekif the initial state at time 0 is x(0) =x; the solution exists andis unique (even if() is discontinuous).

If() is not continuous, as may be the case when () is a modelpredictive control (MPC) law, then f((), ()) may not be continuous.

In this case we assume that f((), ()) is locally bounded.

Definition 1 (Locally bounded)

A function f :X X is locally bounded if, for any x X, there exists aneighborhood N ofxsuch that f(N) is a bounded set, i.e., if there existsa M >0 such that |f(x)| Mfor all x N.


Stability and equilibrium point

We would like to be sure that the controlled system is stable, i.e., that

small perturbations of the initial state do not cause large variations in thesubsequent behavior of the system, and that the state converges to a

desired state or, if this is impossible due to disturbances, to a desired setof states.If convergence to a specified state, x say, is sought, it is desirable for this

state to be an equilibrium point:

Definition 2 (Equilibrium point)

A point x is an equilibrium point ofx+ =f(x) ifx(0) =x impliesx(k) = (k; x) = x for all k 0. Hence x is an equilibrium point if itsatisfies

x =f(x)


Positive invariant set

In other situations, for example when studying the stability properties ofan oscillator, convergence to a specified closed set A Rn is sought.

If convergence to a set A is sought, it is desirable for the set A to bepositive invariant:

Definition 3 (Positive invariant set)

A setA is positive invariant for the system x+ =f(x) ifx A impliesf(x) A.

Clearly, any solution ofx+ =f(x) with initial state in A, remains in A.The (closed) set A = {x} consisting of a (single) equilibrium point is aspecial case;x A (x= x) implies f(x) A(f(x) = x).

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Distance to a set; set addition

Define distance from point x to set A

|x|A := infzA |x z|

IfA = {x}, then |x|A = |x x| which reduces to |x| when x = 0.

Set addition: A B:= {a+b| a A, b B}.


K, K, KL, and PD functions

Definition 4

A function : R0 R0 belongs to class K if it is continuous, zeroat zero, and strictly increasing;

: R0 R0 belongs to class K if it is a class K and unbounded((s) as s ).

A function : R0 I0 R0 belongs to class KL if it iscontinuous and if, for eacht 0, (, t) is a classK function and foreach s 0, (s, ) is nonincreasing and satisfies limt (s, t) = 0.

A function : R R0 belongs to classPD (is positive definite) if itis continuous and positive everywhere except at the origin.


Some useful properties ofKfunctions

The following useful properties of these functions are established in Khalil(2002, Lemma 4.2):

if1() and 2() are K functions (K functions), then 11 () and

(1 2)() :=1(2()) are K functions (K functions).

Moreover, if1() and 2() are K functions and () is a KLfunction, then (r, s) = 1((2(r), s)) is a KL function.


Stability Definitions

Definition 5 (Various forms of stability (constrained))

SupposeXRn

is positive invariant for x+

=f(x), thatA is closed andpositive invariant forx+ =f(x), and that A lies in the interior ofX. ThenAis

1 locally stable in X if, for each >0, there exists a = ()> 0 suchthat x X (A B), implies|(i; x)|A < for all i I0.

a

2 locally attractive inX if there exists a >0 such that

x X (A B) implies|(i; x)|A 0 as i .

3 attractive in X if|(i; x)|A 0 as i for all x X.

aB denotes the unit ball in Rn.



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Stability Definitions (cont.)

Definition 5 (Various forms of stability (constrained))

4 locally asymptotically stable inX if it is locally stable in Xand locally

attractive in X.

5 asymptotically stable with a region of attraction X if it is locallystable in Xand attractive in X.

6 locally exponentially stable with a region of attraction X if there exist >0, c> 0, and (0, 1) such that x X (A B) implies|(i; x)|A c|x|Ai for all i I0.

7 exponentiallystable with a region of attraction X if there exists ac> 0 and a (0, 1) such that |(i; x)|A c|x|A

i for all x X,all i I0.


Asymptotic stability stronger definition

An alternative, stronger definition of asymptotic stability is becomingpopular. Ill call it the KL version.

Definition 6 (Asymptotic stability (constrained KL version))

SupposeX Rn is positive invariant for x+ =f(x), that A is closed and

positive invariant forx+ =f(x), and thatA lies in the interior ofX. ThesetA is asymptotically stable with a region of attraction X forx+ =f(x)if there exists aKL function () such that, for each x X

|(i; x)|A (|x|A , i) i I0 (1)

See Teel and Zaccarian(2006) and the Notes on Recent MPCLiterature link on: www.che.wisc.edu/~jbraw/mpc for furtherdiscussion of the differences in the two definitions.Iff() is continuous, the two definitions are equivalent.


Lyapunov function

Definition 7 (Lyapunov function)

A function V : Rn R0 is said to be a Lyapunov function for the systemx+ =f(x) and setA if there exist functions i K, i= 1, 2 and3 PD such that for any x R

n,

V(x) 1(|x|A) (2)

V(x) 2(|x|A) (3)

V(f(x)) V(x) 3(|x|A) (4)

IfV() satisfies (2)(4) for all x X where X A is a positive invariantset forx+ =f(x), then V() is said to be a Lyapunovfunction in X forthe system x+ =f(x) and set A.


Some fine print

Remark 8

It is shown in Jiang and Wang (2002), Lemma 2.8, that, under the

assumption that f() is continuous, we can always assume that 3() in (4)is aK function.

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Lyapunov function Asymptotic stability

The existence of a Lyapunov function is a sufficient condition for globalasymptotic stability as shown in the next result which we prove under theassumption, common in MPC, that 3() isK function.

Theorem 9 (Lyapunov function and GAS)

Suppose V()is a Lyapunov function for x+ =f(x) and setA with 3()aK function. Then A is globally asymptotically stable.


Converse Lyapunov theorem Asymptotic stability

Theorem9 merely provides a sufficient condition for global asymptoticstability that might be thought to be conservative. The next result, a

converse stability theorem by Jiang and Wang (2002), establishes necessityunder a stronger hypothesis, namely that f() is continuous rather thanlocally bounded.

Theorem 10 (Converse theorem for asymptotic stability)

LetA be compact and f() continuous. Suppose that the setA is globallyasymptotically stable for the system x+ =f(x). Then there exists asmooth Lyapunov function for the system x+ =f(x)and setA.

A proof of this result is given in Jiang and Wang (2002), Theorem 1,part 3.


Asymptotic stability Constrained case

Theorem 11 (Lyapunov function for asymptotic stability (constrainedcase))

Suppose X Rn is positive invariant for x+ =f(x), thatA is closed andpositive invariant for x+ =f(x), and thatA lies in the interior of X. Ifthere exists a Lyapunov function in X for the system x+ =f(x)and setAwith 3()a K function, then A is asymptotically stable for x+ =f(x)with a region of attraction X.


Exponential stability Constrained case

Theorem 12 (Lyapunov function for exponential stability)

Suppose X Rn is positive invariant for x+ =f(x), thatA is closed andpositive invariant for x+ =f(x), and thatA lies in the interior of X. Ifthere exists V : Rn R0 satisfying the following properties for all x X

a1 |x|

A V(x) a2 |x|

A

V(f(x)) V(x) a3 |x|

A

in which a1, a2, a3, >0, thenA is exponentially stable for x+ =f(x)with a region of attraction X.



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Converse theorem for exponential stability

Exercise B.3: A converse theorem for exponential stability

a Assume that the origin is globally exponentially stable (GES) for thesystem

x+ =f(x)

in whichfis Lipschitz continuous. Show that there exists a Lipschitz

continuous Lyapunov function V() for the system satisfying for allx Rn

a1 |x| V(x) a2 |x|

V(f(x)) V(x) a3 |x|

in whicha1, a2, a3, >0.Hint: Consider summing the solution |(i; x)|on ias a candidateLyapunov function V(x).

b Establish also that in the Lyapunov function defined above, any >0

is valid, and the constant a3 can be chosen as large as one wishes.


Robust Stability

We turn now to stability conditions for systems subject to bounded

disturbances (not vanishingly small) and described by

x+ =f(x, w) (5)

where the disturbance w lies in the compact set W.This system may equivalently be described by the difference inclusion

x+ F(x) (6)

where the setF(x) := {f(x, w) | wW}

Let S(x) denote the set of all solutions of (5) or (6) with initial state x.


Positive invariant sets

We require, in the sequel, that the set A is positive invariant for (5) (orforx+ F(x)):

Definition 13 (Positive invariance with disturbances)

The set A is positive invariant for x+ =f(x, w), wW ifx A impliesf(x, w) A for all wW; it is positive invariant for x+ F(x) ifx Aimplies F(x) A.

Clearly the two definitions are equivalent; A is positive invariant forx+ =f(x, w), wW, if and only if it is positive invariant for x+ F(x).In Definitions14-16, we use positive invariant to denote positiveinvariant for x+ =f(x, w), w W or for x+ F(x).


Stability and attraction

Definition 14 (Local stability (disturbances))

The closed positive invariant set A is locally stablefor x+

=f(x, w),wW (or for x+ F(x)) if, for all >0, there exists a >0 such that,for each x satisfying|x|A < , each solution S(x) satisfies|(i)|A < for all i I0.

Definition 15 (Global attraction (disturbances))

The closed positive invariant set A is globally attractivefor the systemx+ =f(x, w), wW(or for x+ F(x)) if, for each x Rn, eachsolution() S(x) satisfies|(i)|A 0 as i .



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Asymptotic stability

Definition 16 (GAS (disturbances))

The closed positive invariant set A is globally asymptotically stablefor

x+

=f(x, w), wW(or for x+

F(x)) if it is locally stable and globallyattractive.

An alternative definition of global asymptotic stability ofA forx+ =f(x, w), wW, ifA is compact, is the existence of aKL function() such that for each x Rn, each S(x) satisfies|(i)|A (|x|A, i) for all i I0.


Lyapunov function

To cope with disturbances we require a modified definition of a Lyapunovfunction.

Definition 17 (Lyapunov function (disturbances))

A function V :Rn

R0 is said to be a Lyapunov function for the systemx+ =f(x, w), wW(or for x+ F(x)) and set A if there exist

functions i K, i= 1, 2 and 3 PD such that for any x Rn,

V(x) 1(|x|A) (7)

V(x) 2(|x|A) (8)

supzF(x)

V(z) V(x) 3(|x|A) (9)


GAS (disturbances)

Inequality9 ensures V(f(x, w)) V(x) 3(|x|A) for all wW. The

existence of a Lyapunov function for the system x+

F(x) and set A is asufficient condition for A to be globally asymptotically stable forx+ F(x) as shown in the next result.

Theorem 18 (Lyapunov function for GAS (disturbances))

Suppose V()is a Lyapunov function for x+ =f(x, w), wW(or forx+ F(x)) and setA with 3() a K function. ThenA is globallyasymptotically stable for x+ =f(x, w), w W(or for x+ F(x)).


Input-to-State Stability

In input-to-state stability (Sontag and Wang, 1995; Jiang and Wang,2001) we seek a bound on the state in terms of a uniform bound on thedisturbancesequence w := {w(0), w(1), . . .}. Let denote the usual

norm for sequences, i.e., w := supk0 |w(k)|.

Definition 19 (Input-to-state stable (ISS))

The system x+ =f(x, w) is (globally) input-to-state stable (ISS) if there

exists aKL function () and aK function () such that, for eachx Rn, and each disturbance sequence w = {w(0), w(1), . . .} in

|(i; x, wi)| (|x| , i) +(wi)

for alli I0, where (i; x, wi) is the solution, at time i, if the initial state

is xat time 0 and the input sequence is wi := {w(0), w(1), . . . , w(i 1)}.



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The basic nonlinear, constrained MPC problem

The system model isx+ =f(x, u) (10)

Both state and input are subject to constraints

x(k) X , u(k) U for allk I0

Given an integerN(referred to as the finite horizon), and an inputsequence u of lengthN, u = {u(0), u(1), . . . , u(N 1)}, let (k; x, u)denote the solution of (10) at timekfor a given initial statex(0) =x.

Terminal constraint (and penalty)

(N; x, u) Xf X


Feasible sets

The set of feasible initial states and associated control sequences

ZN = {(x, u) | u(k) U, (k; x, u) X for all k I0:N1,

and (N; x, u) Xf}

and Xf X is the feasible terminal set.

The set of feasible initial states is

XN= {x Rn | u UN such that (x, u) ZN} (11)

For each x XN, the corresponding set of feasible input sequences is

UN(x) = {u | (x, u) ZN}


Cost function and control problem

For any state x Rn and input sequence u UN, we define

VN(x, u) =

N1k=0

((k; x, u), u(k)) +Vf((N; x, u))

(x, u) is the stage cost; Vf(x(N)) is the terminal cost

Consider the finite horizon optimal control problem

PN(x) : minuUN

VN(x, u)


Control law and closed-loop system

The control law is

N(x) = u0(0; x)

The optimum may not be unique; then N() is a point-to-set map

Closed-loop system

x+ =f(x, N(x)) difference equation

x+ f(x, N(x)) difference inclusion

Nominal closed-loop stability question; is the origin stable?

If yes, what is the region of attraction? All ofXN?



9/12

Inherent robustness of the nominal controller

Consider a process disturbance d,x+ =f(x, (x)) +d

A measurement disturbancexm = x+ e

Nominal controller with disturbance

x+ f(x, N(xm)) +d

x+ f(x, N(x+ e)) +d

x+ F(x, w) w = (d, e)

Robust stability; is the system x+ F(x, w) input-to-state stableconsidering w= (d, e) as the input.


Basic MPC assumptions

Assumption 20 (Continuity of system and cost)

The functions f : Rn Rm Rn, : Rn Rm R0 and

Vf : Rn

R0 are continuous, f(0, 0) = 0, (0, 0) = 0, and Vf(0) = 0.

Assumption21 (Properties of constraint sets)

The set U is compact and contains the origin. The sets X and Xf are

closed and contain the origin in their interiors, Xf X.


Basic MPC assumptions

Assumption 22 (Basic stability assumption)

For any x Xfthere exists u:= f(x) Usuch that f(x, u) Xf andVf(f(x, u)) + (x, u) Vf(x).

Note: understanding this requirement created a big research challenge forthe development of nonlinear MPC. Credit the celebrated quasi-infinitehorizon work of Chen and Allgower (1998) for cracking this problem.

Assumption23 (Bounds on stage and terminal costs)

The stage cost () and the terminal cost Vf() satisfy

(x, u) 1(|x|) x XN, u U

Vf(x) 2(|x|) x Xf

in which 1() and 2() are K functions


Optimal MPC cost function as Lyapunov function

We show that the optimal cost V0N() is a Lyapunov function for theclosed-loop system. We require three properties.

Lower bound.

V0N(x) 1(|x|) for all x XN

Given the definition ofVN(x, u) as a sum of stage costs, we have using

Assumption23

VN(x, u) (x, u(0; x)) 1(|x|) for all x XN, u UN

so the first property is established.



10/12

MPC cost function as Lyapunov function cost decrease

Next we require the cost decrease

V0N(f(x, N(x)) V0N(x) 3(|x|) for all x XN

At state x XN, consider the optimal sequenceu0(x) = {u(0; x), u(1; x), . . . , u(N 1; x)}, and generate a candidate

sequencefor the successor state,x

+

:=f(x, N(x))

u= {u(1; x), u(2; x), . . . , u(N 1; x), f(x(N))}

withx(N) :=(N; x, u). This candidate is feasible forx+ because Xf iscontrol invariant under control lawf() (Assumption22).The cost is

VN(x+, u) = V0N(x) (x, u(0; x))

Vf(x(N)) +(x(N), f(x(N))) +Vf(f(x(N), f(x(N))))


Cost decrease (cont.)

But by Assumption22

Vf(f(x, f(x))) +(x, f(x)) Vf(x) for all x Xf

so we have that

VN(x+, u) V0N(x) (x, u(0; x))

The optimal cost is certainly no worse, giving

V0N(x+) V0N(x) (x, u(0; x))

V0N(x+) V0N(x) 1(|x|) for all x XN

which is the desired cost decrease with the choice 3() = 1().


Upper bound

Finally we require the upper bound.

V0N(x) 2(|x|) for all x XN

Surprisingly, this one turns out to be the most involved.First, we have the bound from Assumption23

Vf(x) 2(|x|) for all x Xf

Next we show that V0N(x) Vf(x) forx Xf,N 1.Consider N= 1,

V01(x) = minuU{(x, u) +Vf(f(x, u)) | f(x, u) Xf}

Vf(x) x Xf


Dynamic programming recursion

Next considerN= 2, and optimal control law 2()

V02(x) = minuU{(x, u) +V01(f(x, u)) | f(x, u) X1} x X2

=(x, 2(x)) +V01(f(x, 2(x))) x X2

(x, 1(x)) +V0

1(f(x, 1(x))) x X1

(x, 1(x)) +Vf(f(x, 1(x))) x X1

=V01(x) x X1

Therefore

V02(x) Vf(x) x Xf

Continuing this recursion gives for all N 1

V0N(x) Vf(x) x Xf



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Extending the upper bound from Xf to XN

Question: When can we extend this bound from Xf to the (possiblyunbounded!) set XN? Recall that V

0N() is not necessarily continuous.

Answer: A function can be upper bounded by a K function if andonly if it is locally bounded.2

We know from continuity off() (Assumption20) that VN(x, u) is acontinuous function, hence locally bounded, and therefore so isV0N(x).

Therefore, there exists () K such that

V0N(x) (|x|) for all x XN

Be aware that the MPC literature has been confused about therequirements for this last result.

2See Proposition 10 of Notes on Recent MPC Literature link on:

www.che.wisc.edu/~jbraw/mpc. Thanks also to Andy Teel.SOCN 2013 MPC short course 41 / 45

Asymptotic stability of constrained nonlinear MPC

Why you want a Lyapunov function

We have established that the optimal cost V0N() is a Lyapunovfunction on XNfor the closed-loop system.

Therefore, the origin is asymptotically stable (KLversion) with region

of attraction XN.We can also establish robust stability, but lets do that later.

If we strengthen the properties of(), we can strengthen theconclusion to exponential stability.

Notice the essential role that V0N() plays in the stability analysis ofMPC.

In economic MPC we lose this Lyapunov function and have to work tobring it back.


Recommended exercises

Stability definitions. Example 2.3

Lyapunov functions. Exercise B.2B.4.4

Dynamic programming. Exercise C.1C.2.4

MPC stability results. Theorem 7 and Example 1.3

Exercises 2.11, 2.14, 2.154

3Notes on Recent MPC Literature link on: www.che.wisc.edu/~jbraw/mpc.4Rawlings and Mayne (2009, Appendices B and C). Downloadable from

www.che.wisc.edu/~jbraw/mpc.SOCN 2013 MPC short course 43 / 45

Further Reading I

H. Chen and F. Allgower. A quasi-infinite horizon nonlinear model

predictive control scheme with guaranteed stability. Automatica, 34(10):12051217, 1998.

Z.-P. Jiang and Y. Wang. Input-to-state stability for discrete-timenonlinear systems. Automatica, 37:857869, 2001.

Z.-P. Jiang and Y. Wang. A converse Lyapunov theorem for discrete-time

systems with disturbances. Sys. Cont. Let., 45:4958, 2002.

H. K. Khalil. Nonlinear Systems. Prentice-Hall, Upper Saddle River, NJ,third edition, 2002.

J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory andDesign. Nob Hill Publishing, Madison, WI, 2009. 576 pages, ISBN

978-0-9759377-0-9.

E. D. Sontag and Y. Wang. On the characterization of the input to statestability property. Sys. Cont. Let., 24:351359, 1995.

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Further Reading II

A. R. Teel and L. Zaccarian. On uniformity in definitions of global

asymptotic stability for time-varying nonlinear systems. Automatica, 42:22192222, 2006.


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