Discrete Time Systems

In this book, we investigate model predictive control for discrete time nonlinear control systems of the form

x⁺=f (x, u). (2.1)

Here, the transition mapf :X×U→Xassigns the statex⁺∈Xat the next time instant to each pair of statex∈Xand control valueu∈U. The state spaceX and the control value spaceU are arbitrary metric spaces, i.e., sets in which we can measure distances between two elementsx, y∈Xoru, v∈U by metricsd_X(x, y) or dU(u, v), respectively. Readers less familiar with metric spaces may think of X=R^dandU=R^mford, m∈Nwith the Euclidean metricsd_X(x, y)= x−y anddU(u, v)= u−vinduced by the usual Euclidean norm·, although some of our examples use different spaces. While most of the systems we consider possess continuous transition mapsf, we do not require continuity in general.

The set of finite control sequencesu(0), . . . , u(N−1)forN∈Nwill be denoted byU^N and the set of infinite control sequencesu(0), u(1), u(2), . . .byU^∞. Note that we may interpret the control sequences as functionsu: {0, . . . , N−1} →Uor u:N0→U, respectively. For either type of control sequences we will briefly write u(·)or simplyuif there is no ambiguity. WithN_∞we denote the natural numbers including∞and withN0the natural numbers including 0.

A trajectory of (2.1) is obtained as follows: given an initial valuex₀∈Xand a control sequenceu(·)∈U^K forK∈N∞, we define the trajectoryxu(k)iteratively via

x_u(0)=x0, x_u(k+1)=f

x_u(k), u(k)

, (2.2)

for allk∈N0ifK= ∞and fork=0,1, . . . , K−1 otherwise. Whenever we want to emphasize the dependence on the initial value we writex_u(k, x₀).

An important basic property of the trajectories is the cocycle property: given an initial valuex₀∈X, a controlu∈U^Nand time instantsk₁, k₂∈ {0, . . . , N−1}with k1≤k2the solution trajectory satisfies

x_u(k₂, x₀)=x_u(·+k1)

k₂−k₁, x_u(k₁, x₀)

. (2.3)

L. Grüne, J. Pannek, Nonlinear Model Predictive Control, Communications and Control Engineering,

DOI10.1007/978-0-85729-501-9_2, © Springer-Verlag London Limited 2011

13

14 2 Discrete Time and Sampled Data Systems Here, the shifted control sequenceu(· +k₁)∈U^N−k1 is given by

u(· +k1)(k):=u(k+k1), k∈ {0, . . . , N−k1−1}, (2.4) i.e., if the sequenceu consists of theN elementsu(0), u(1), . . . , u(N −1), then the sequenceu˜=u(· +k₁)consists of theN−k₁elementsu(0)˜ =u(k₁),u(1)˜ = u(k1+1), . . . ,u(N˜ −k1−1)=u(N−1). With this definition, the identity (2.3) is easily proved by induction using (2.2).

We illustrate our class of models by three simple examples—the first two being in fact linear.

Example 2.1 One of the simplest examples of a control system of type (2.1) is given byX=U=Rand

x⁺=x+u=:f (x, u).

This system can be interpreted as a very simple model of a vehicle on an infinite straight road in whichu∈Ris the traveled distance in the period until the next time instant. Foru >0 the vehicle moves right and foru <0 it moves left.

Example 2.2 A slightly more involved version of Example2.1is obtained if we consider the statex=(x1, x2)∈X=R², wherex1represents the position andx2

the velocity of the vehicle. With the dynamics x₁⁺

x₂⁺

=

x₁+x₂+u/2 x₂+u

=:f (x, u)

on an appropriate time scale the controlu∈U=Rcan be interpreted as the (con-stant) acceleration in the period until the next time instant. For a formal derivation of this model from a continuous time system, see Example2.6, below.

Example 2.3 Another variant of Example2.1is obtained if we consider the vehicle on a road which forms an ellipse, cf. Fig.2.1, in which half of the ellipse is shown.

Here, the set of possible states is given by X=

x∈R²

x1

2x2 =1 . Fig. 2.1 Illustration of

Example2.3

2.1 Discrete Time Systems 15 Since X is a compact subset of R² (more precisely a submanifold, but we will not need this particular geometric structure) we can use the metric induced by the Euclidean norm onR², i.e.,d_X(x, y)= x−y. Defining the dynamics

x₁⁺ x₂⁺

=

sin(ϑ (x)+u) cos(ϑ (x)+u)/2

=:f (x, u)

withu∈U=Rand

ϑ (x)=

arccos 2x2, x1≥0, 2π−arccos 2x2, x1<0

the vehicle moves on the ellipse with traveled distanceu∈U=Rin the next time step, where the traveled distance is now expressed in terms of the angleϑ. Foru >0 the vehicle moves clockwise and foru <0 it moves counterclockwise.

The main purpose of these very simple examples is to provide test cases which we will use in order to illustrate various effects in model predictive control. Due to their simplicity we can intuitively guess what a reasonable controller should do and often even analytically compute different optimal controllers. This enables us to compare the behavior of the NMPC controller with our intuition and other controllers. More sophisticated models will be introduced in the next section.

As outlined in the introduction, the model (2.1) will serve for generating the predictionsxu(k, x(n))which we need in the optimization algorithm of our NMPC scheme, i.e., (2.1) will play the role of the model (1.1) used in the introduction.

Clearly, in general we cannot expect that this mathematical model produces exact predictions for the trajectories of the real process to be controlled. Nevertheless, during Chaps. 3–7 and in Sects. 8.1–8.4 of this book we will suppose this idealized assumption. In other words, given the NMPC-feedback lawμ:X→U, we assume that the resulting closed-loop system satisfies

x⁺=f

x, μ(x)

(2.5) withf from (2.1). We will refer to (2.5) as the nominal closed-loop system.

There are several good reasons for using this idealized assumption: First, satis-factory behavior of the nominal NMPC closed loop is a natural necessary condition for the correctness of our controller—if we cannot ensure proper functioning in the absence of modeling errors we can hardly expect the method to work under real life conditions. Second, the assumption that the prediction is based on an exact model of the process considerably simplifies the analysis and thus allows us to derive suf-ficient conditions under which NMPC works in a simplified setting. Last, based on these conditions for the nominal model (2.5), we can investigate additional robust-ness conditions which ensure satisfactory performance also for the realistic case in which (2.5) is only an approximate model for the real closed-loop behavior. This issue will be treated in Sects. 8.5–8.9.

16 2 Discrete Time and Sampled Data Systems