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Preliminaries for Model Predictive Control course

James B. Rawlings

Department of Chemical and Biological EngineeringUniversity of Wisconsin–Madison

Insitut für Systemtheorie und RegelungstechnikUniversität StuttgartStuttgart, Germany

June 20, 2011

Stuttgart – June 2011 MPC short course 1 / 46

Outline

1 Introduction

2 Stability, equilibria, invariant sets

3 Lyapunov function theory

4 Disturbances and robust stability

5 Basic MPC problem and nominal stability


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Predictive control

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Measurement

MH EstimateMPC control

Forecast

t time

Reconcile the past Forecast the future

sensorsy

actuatorsu

minu (t )

T 0|y sp − g (x , u )|

2Q

+ |u sp − u |2R

dt

ẋ = f (x , u )

x (0) = x 0 (given)

y = g (x , u )Stuttgart – June 2011 MPC short course 3 / 46

State estimation

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0 00 01 11 1 0 00 00 01 11 11 1

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000111

Measurement

MH EstimateMPC control

Forecast

t time

Reconcile the past Forecast the future

sensorsy

actuatorsu

minx 0,w (t )

0−T

|y − g (x , u )|2R + |ẋ − f (x , u )|2Q dt

ẋ = 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 formulation x and u denote, respectively, the current state andcontrol, and x + the successor state.

We assume in the sequel that the function f : Rn × Rm → Rn iscontinuous.

Letφ(k ; x , u)

denote the solution of x + = f (x , u ) at time k if the initial state is

x (0) = x and 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.


Existence of solutions to model

If a state-feedback control law u = κ(x ) has been chosen, theclosed-loop system is described by x + = f (x , κ(x )).

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

If κ(·) is not continuous, as may be the case when κ(·) is a model

predictive control (MPC) law, then f ((·), κ(·)) may not be continuous.In this case we assume that f ((·), κ(·)) is locally bounded .

Definition 1 (Locally bounded)

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


http://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpc

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Stability and equilibrium point

We would like to be sure that the controlled system is “stable”, i.e., thatsmall perturbations of the initial state do not cause large variations in thesubsequent behavior of the system, and that the state converges to adesired 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 thisstate to be an equilibrium point:

Definition 2 (Equilibrium point)

A point x ∗ is an equilibrium point of x + = f (x ) if x (0) = x ∗ impliesx (k ) = φ(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 of an 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 set A is positive invariant for the system x + = f (x ) if x ∈ A impliesf (x ) ∈ A.

Clearly, any solution of x + = 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 := inf z ∈A|x − z |

If A = {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 σ : R≥0 → R≥0 belongs to class K if it is continuous, zeroat zero, and strictly increasing;

σ : R≥0 → R≥0 belongs to class K∞ if it is a class K and unbounded(σ(s )→∞ as s → ∞).

A function β : R≥0 × I≥0 → R≥0 belongs to class KL if it iscontinuous and if, for each t ≥ 0, β (·, t ) is a class K function and foreach s ≥ 0, β (s , ·) is nonincreasing and satisfies limt →∞ β (s , t ) = 0.

A function γ : R→ R≥0 belongs to class PD (is positive definite) if itis continuous and positive everywhere except at the origin.


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Some useful properties of K functions

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

if α1(·) and α2(·) are K functions (K∞ functions), then α−11 (·) and

(α1 ◦ α2)(·) := α1(α2(·)) are K functions (K∞ functions).

Moreover, if α1(·) and α2(·) are K functions and β (·) is a KLfunction, then σ(r , s ) = α1(β (α2(r ), s )) is a KL function.


Stability — Definitions

Definition 5 (Various forms of stability (constrained))

Suppose X ⊂ Rn is positive invariant for x + = f (x ), that A is closed andpositive invariant for x + = f (x ), and that A lies in the interior of X . ThenA is

1 locally stable in X if, for each ε > 0, there exists a δ = δ (ε) > 0 such

that x ∈ X ∩ (A⊕ δ B ), implies |φ(i ; x )|A < ε for all i ∈ I≥0.a

2 locally attractive in X if there exists a η > 0 such thatx ∈ X ∩ (A⊕ ηB ) implies |φ(i ; x )|A → 0 as i → ∞.

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

aB denotes the unit ball in Rn.


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Lyapunov function

Definition 7 (Lyapunov function)

A function V : Rn → R≥0 is said to be a Lyapunov function for the system

x +

= f (x ) and set A if there exist functions αi ∈ K∞, i = 1, 2 andα3 ∈ PD such that for any x ∈ Rn,

V (x ) ≥ α1(|x |A) (2)

V (x ) ≤ α2(|x |A) (3)

V (f (x ))− V (x ) ≤ −α3(|x |A) (4)

If V (·) satisfies (2)–(4) for all x ∈ X where X ⊃ A is a positive invariantset for x + = f (x ), then V (·) is said to be a Lyapunov function 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 theassumption that f (·) is continuous, we can always assume that α

3(·) in (4)

is a K∞ function.


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Lyapunov function — Asymptotic stability

The existence of a Lyapunov function is a sufficient condition for global

asymptotic stability as shown in the next result which we prove under theassumption, common in MPC, that α3(·) is K∞ function.

Theorem 9 (Lyapunov function and GAS)

Suppose V (·) is a Lyapunov function for x + = f (x ) and set A with α3(·)a K∞ function. Then A is globally asymptotically stable.


Converse Lyapunov theorem — Asymptotic stability

Theorem 9 merely provides a sufficient condition for global asymptoticstability that might be thought to be conservative. The next result, aconverse 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)

Let A be compact and f (·) continuous. Suppose that the set A is globally asymptotically stable for the system x + = f (x ). Then there exists asmooth Lyapunov function for the system x + = f (x ) and set A.

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


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Asymptotic stability — Constrained case

Theorem 11 (Lyapunov function for asymptotic stability (constrained

case))

Suppose X ⊂ Rn is positive invariant for x + = f (x ), that A is closed and positive invariant for x + = f (x ), and that A lies in the interior of X. If there exists a Lyapunov function in X for the system x + = f (x ) and set Awith α3(·) 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 ), that A is closed and positive invariant for x + = f (x ), and that A lies in the interior of X. If there exists V : Rn → R≥0 satisfying the following properties for all x ∈ X

a1 |x |σA ≤ V (x ) ≤ a2 |x |σAV (f (x ))− V (x ) ≤ −a3 |x |

σ

A

in which a1, a2, a3, σ > 0, then A 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 which f is Lipschitz continuous. Show that there exists a Lipschitzcontinuous Lyapunov function V (·) for the system satisfying for allx ∈ Rn

a1 |x |σ ≤ V (x ) ≤ a2 |x |

σ

V (f (x ))− V (x ) ≤ −a3 |x |σ

in which a1, a2, a3, σ > 0.Hint: Consider summing the solution |φ(i ; x )| on i as a candidateLyapunov function V (x ).

b Establish also that in the Lyapunov function defined above, any σ > 0is 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 boundeddisturbances (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 ) | w ∈W}

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


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Positive invariant sets

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

Definition 13 (Positive invariance with disturbances)

The set A is positive invariant for x + = f (x , w ), w ∈W if x ∈ A impliesf (x , w ) ∈ A for all w ∈W; it is positive invariant for x + ∈ F (x ) if x ∈ Aimplies F (x ) ⊆ A.

Clearly the two definitions are equivalent; A is positive invariant forx + = f (x , w ), w ∈W, if and only if it is positive invariant for x + ∈ F (x ).In Definitions 14-16, we use “positive invariant” to denote “positive

invariant 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 stable for x + = f (x , w ),w ∈W (or for x + ∈ F (x )) if, for all ε > 0, there exists a δ > 0 such that,for each x satisfying |x |A < δ , each solution φ ∈ S (x ) satisfies |φ(i )|A < εfor all i ∈ I≥0.

Definition 15 (Global attraction (disturbances))

The closed positive invariant set A is globally attractive for the systemx + = f (x , w ), w ∈W (or for x + ∈ F (x )) if, for each x ∈ Rn, eachsolution φ(·) ∈ S (x ) satisfies |φ(i )|A → 0 as i → ∞.


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Asymptotic stability

Definition 16 (GAS (disturbances))

The closed positive invariant set A is globally asymptotically stable forx + = f (x , w ), w ∈W (or for x + ∈ F (x )) if it is locally stable and globallyattractive.

An alternative definition of global asymptotic stability of A forx + = f (x , w ), w ∈W, if A is compact, is the existence of a KL functionβ (·) such that for each x ∈ Rn, each φ ∈ S (x ) satisfies|φ(i )|A ≤ β (|x |A, i ) for all i ∈ I≥0.


Lyapunov function

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

Definition 17 (Lyapunov function (disturbances))

A function V : Rn → R≥0 is said to be a Lyapunov function for the systemx + = f (x , w ), w ∈W (or for x + ∈ F (x )) and set A if there exist

functions αi ∈ K∞, i = 1, 2 and α3 ∈ PD such that for any x ∈ Rn

,

V (x ) ≥ α1(|x |A) (7)

V (x ) ≤ α2(|x |A) (8)

supz ∈F (x )

V (z )− V (x ) ≤ −α3(|x |A) (9)


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GAS (disturbances)

Inequality 9 ensures V (f (x , w ))− V (x ) ≤ −α3(|x |A) for all w ∈W. Theexistence of a Lyapunov function for the system x + ∈ F (x ) and set A is a

sufficient 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 ), w ∈W (or for x + ∈ F (x )) and set A with α3(·) a K∞ function. Then A is globally asymptotically 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 thedisturbance sequence w := {w (0), w (1), . . .}. Let · denote the usual ∞norm for sequences, i.e., w := supk ≥0 |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 a KL function β (·) and a K function σ(·) such that, for eachx ∈ Rn, and each disturbance sequence w = {w (0), w (1), . . .} in ∞

|φ(i ; x , wi )| ≤ β (|x | , i ) + σ(wi )

for all i ∈ I≥0, where φ(i ; x , wi ) is the solution, at time i , if the initial stateis x at 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 all k ∈ I≥0

Given an integer N (referred to as the finite horizon), and an inputsequence u of length N , u = {u (0), u (1), . . . , u (N − 1)}, let φ(k ; x , u)denote the solution of (10) at time k for a given initial state x (0) = x .

Terminal constraint (and penalty)

φ(N ; x , u) ∈ Xf ⊆ X


Feasible sets

The set of feasible initial states and associated control sequences

ZN = {(x , u) | u (k ) ∈ U, φ(k ; x , u) ∈ X for all k ∈ I0:N −1,

and φ(N ; x , u) ∈ Xf }

and Xf ⊆ X is the feasible terminal set.

The set of feasible initial states is

X N = {x ∈ Rn | ∃u ∈ UN such that (x , u) ∈ ZN } (11)

For each x ∈ X N , the corresponding set of feasible input sequences is

U N (x ) = {u | (x , u) ∈ ZN }


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Cost function and control problem

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

V N (x , u) =N −1k =0

(φ(k ; x , u), u (k )) + V f (φ(N ; x , u))

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

Consider the finite horizon optimal control problem

PN (x ) : minu∈U N

V N (x , u)


Control law and closed-loop system

The control law isκN (x ) = u

0(0; x )

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

Closed-loop system

x +

= f (x , κN (x )) difference equation

x + ∈ f (x , κN (x )) difference inclusion

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

If yes, what is the region of attraction? All of X N ?


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Inherent robustness of the nominal controller

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

A measurement disturbance x m = x + e

Nominal controller with disturbance

x + ∈ f (x , κN (x m)) + d

x + ∈ f (x , κN (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 → R≥0 andV f : R

n → R≥0 are continuous, f (0, 0) = 0, (0, 0) = 0, and V f (0) = 0.

Assumption 21 (Properties of constraint sets)

The set U is compact and contains the origin. The sets X and Xf areclosed and contain the origin in their interiors, Xf ⊆ X.


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Basic MPC assumptions

Assumption 22 (Basic stability assumption)

For any x ∈ Xf there exists u := κf (x ) ∈ U such that f (x , u ) ∈ Xf andV f (f (x , u )) + (x , u ) ≤ V f (x ).

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

Assumption 23 (Bounds on stage and terminal costs)

The stage cost (·) and the terminal cost V f (·) satisfy

(x , u ) ≥ α1(|x |) ∀x ∈ X N , ∀u ∈ U

V f (x ) ≤ α2(|x |) ∀x ∈ Xf

in which α1(·) and α2(·) are K∞ functions


Optimal MPC cost function as Lyapunov function

We show that the optimal cost V 0N (·) is a Lyapunov function for theclosed-loop system. We require three properties.Lower bound.

V 0N (x ) ≥ α1(|x |) for all x ∈ X N

Given the definition of V N (x , u) as a sum of stage costs, we have usingAssumption 23

V N (x , u) ≥ (x , u (0; x )) ≥ α1(|x |) for all x ∈ X N , u ∈ UN

so the first property is established.


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MPC cost function as Lyapunov function – cost decrease

Next we require the cost decrease

V 0N (f (x , κN (x )) ≤ V 0N (x )− α3(|x |) for all x ∈ X N

At state x ∈ X N , consider the optimal sequence

u0(x ) = {u (0; x ), u (1; x ), . . . , u (N − 1; x )}, and generate a candidate sequence for the successor state, x + := f (x , κN (x ))

ũ = {u (1; x ), u (2; x ), . . . , u (N − 1; x ), κf (x (N ))}

with x (N ) := φ(N ; x , u). This candidate is feasible for x + because Xf iscontrol invariant under control law κf (·) (Assumption 22).The cost is

V N (x +, ũ) = V 0N (x )− (x , u (0; x ))

− V f (x (N )) + (x (N ), κf (x (N ))) + V f (f (x (N ), κf (x (N ))))


Cost decrease (cont.)

But by Assumption 22

V f (f (x , κf (x ))) + (x , κf (x )) ≤ V f (x ) for all x ∈ Xf

so we have that

V N (x +, ũ) ≤ V 0N (x )− (x , u (0; x ))

The optimal cost is certainly no worse, giving

V 0N (x +) ≤ V 0N (x )− (x , u (0; x ))

V 0N (x +) ≤ V 0N (x )− α1(|x |) for all x ∈ X N

which is the desired cost decrease with the choice α3(·) = α1(·).


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Upper bound

Finally we require the upper bound.

V 0N (x ) ≤ α2(|x |) for all x ∈ X N

Surprisingly, this one turns out to be the most involved.First, we have the bound from Assumption 23

V f (x ) ≤ α2(|x |) for all x ∈ Xf

Next we show that V 0N (x ) ≤ V f (x ) for x ∈ Xf , N ≥ 1.Consider N = 1,

V 01 (x ) = minu ∈U{(x , u ) + V f (f (x , u )) | f (x , u ) ∈ Xf }

≤ V f (x ) x ∈ Xf


Dynamic programming recursion

Next consider N = 2, and optimal control law κ2(·)

V 02 (x ) = minu ∈U{(x , u ) + V 01 (f (x , u )) | f (x , u ) ∈ X 1} x ∈ X 2

= (x , κ2(x )) + V 0

1 (f (x , κ2(x ))) x ∈ X 2

≤ (x , κ1(x )) + V 0

1 (f (x , κ1(x ))) x ∈ X 1

≤ (x , κ1(x )) + V f (f (x , κ1(x ))) x ∈ X 1

= V 0

1 (x ) x ∈ X 1

ThereforeV 02 (x ) ≤ V f (x ) x ∈ Xf

Continuing this recursion gives for all N ≥ 1

V 0N (x ) ≤ V f (x ) x ∈ Xf


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Extending the upper bound from Xf to X N

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

0N (·) is not necessarily continuous.

Answer: A function can be upper bounded by a K∞ function if and

only if it is locally bounded.2

We know from continuity of f (·) (Assumption 20) that V N (x , u) is acontinuous function, hence locally bounded, and therefore so isV 0N (x ).Therefore, there exists β (·) ∈ K∞ such that

V 0N (x ) ≤ β (|x |) for all x ∈ X N

Be aware that the MPC literature has been confused about the

requirements 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.


Asymptotic stability of constrained nonlinear MPC

Why you want a Lyapunov function

We have established that the optimal cost V 0N (·) is a Lyapunovfunction on X N for the closed-loop system.

Therefore, the origin is asymptotically stable (KL version) with regionof attraction X N .

We can also establish robust stability, but let’s do that later.

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

Notice the essential role that V 0N (·) plays in the stability analysis of MPC.

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


http://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpc

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Recommended exercises

Stability definitions. Example 2.3

Lyapunov functions. Exercise B.2–B.4.4

Dynamic programming. Exercise C.1–C.2.4

MPC stability results. Theorem 7 and Example 1.3

Exercises 2.11, 2.14, 2.154

3“Notes 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.Stuttgart – June 2011 MPC short course 43 / 46

Further Reading I

H. Chen and F. Allgöwer. A quasi-infinite horizon nonlinear modelpredictive control scheme with guaranteed stability. Automatica, 34(10):1205–1217, 1998.

Z.-P. Jiang and Y. Wang. Input-to-state stability for discrete-timenonlinear systems. Automatica, 37:857–869, 2001.

Z.-P. Jiang and Y. Wang. A converse Lyapunov theorem for discrete-timesystems with disturbances. Sys. Cont. Let., 45:49–58, 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 and Design. Nob Hill Publishing, Madison, WI, 2009. 576 pages, ISBN978-0-9759377-0-9.

E. D. Sontag and Y. Wang. On the characterization of the input to statestability property. Sys. Cont. Let., 24:351–359, 1995.


http://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpchttp://www.che.wisc.edu/~jbraw/mpc

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Further Reading II

A. R. Teel and L. Zaccarian. On “uniformity” in definitions of globalasymptotic stability for time-varying nonlinear systems. Automatica, 42:2219–2222, 2006.


Acknowledgments

The author is indebted to Luo Ji of the University of Wisconsin andGanzhou Wang of the Universität Stuttgart for their help in organizing thematerial and preparing the overheads.

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