4.2 Preliminaries
4.2.1 Invariant Sets
Given a set Ω and an initial state x0 ∈Ω⊆X for a dynamic system, it is of interest to determine whether the unfolding states of the system, xk+1 = f(xk, wk), whose solution is given as φk(xk, wk) remains inside the set X for all time.
Definition 4.1 (Positive Invariant set) The set Ω ⊂ Rn is positively invariant for the system xk+1 =f(xk) iff ∀x0 ∈Ωthe system states satisfy xk ∈Ω, ∀k ∈Z+. Definition 4.2 (Robust Positively Invariant set) [90] The set Ω ⊂ Rn is ro- bust positively invariant (RPI) for the system xk+1 = f(xk, wk) iff ∀x0 ∈ Ω the system states satisfy xk ∈Ω, ∀wk ∈W, ∀k ∈Z+.
The following are the immediate results of the above definition.
Proposition 4.1 The union of two or more RPI sets is also an RPI set.
Remark 4.1 A similar statement as above of RPI set cannot be made about the intersection of RPI sets, even for the case when the disturbance is absent.
Definition 4.3 (Robust Control Invariant set) For a system of the form (4.1), a set Ω⊂Rn is robust control invariant (RCI) set, iff ∀x0 ∈Ω there exits a control input (or a feedback control law, uk :=κ(xk)) uk ∈U such that xk+1 =f(xk, uk)⊕ W∈Ω.
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Definition 4.4 (Maximal Robust Control Invariant set) [90] The set C∞(Ω) is the maximal robust control invariant (MRCI) set contained inΩ for the system of the form (4.1) iff C∞(Ω) is RCI and contains all the RCI sets contained in Ω.
For the RCI set of the system (4.1) and the bounded disturbance W, any control law κ(xk) : Ω →U, given as
κ(xk) = {uk∈U|f(xk, uk)⊕W⊆ Ω, ∀xk ∈Ω} (4.4) ensures the construction of the closed-loop system such that xk+1 = f(xk, κ(xk))⊕ W⊆Ω, ∀xk ∈Ω.
Definition 4.5 (Constraint-admissible set) The set Ω ⊂ Rn is a constraint- admissible set if it is contained in X⊂Rn.
Remark 4.2 The set Ω is a constraint-admissible RPI set if Ω is RPI and is con- tained in X.
There are two other important positive invariant sets, which are worth mention- ing and will be useful in the subsequent discussions. They areminimal-robust positive invariant set andmaximal-robust positive invariant set. The following definitions are consistent with the references [90, 127].
Definition 4.6 (m-RPI set) [127] A constraint-admissible RPI set, X∞, of the system xk+1 =f(xk, wk) is called its minimal robust positive invariant (m-RPI) set if it is contained in every closed RPI set of the system.
Definition 4.7 (M-RPI set) [127] A constraint-admissible RPI set, O∞, of the system xk+1 =f(xk, wk) is called its maximal robust positive invariant (M-RPI) set, if it contains every constraint-admissible RPI set of the system.
Furthermore, it is of natural interest to determine which subset of the given set is compatible with the input constraints.
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Definition 4.8 (Input admissible set) For a given control law uk = κ(xk), the input admissible set (i.e., subset) of Ω⊆Rn is given by,
X ,{xk ∈Ω|κ(xk)∈U}.
It should be noted that if the input constraint U is given as a hyper-rectangle and the control law is given by an appropriate saturation function, sat(·), such that uk =sat(κk(xk)), as is often the case in most practical processes, then X = Ω.
Definition 4.9 (Feasible Input Set) For a givenxk, that satisfies eqns. (4.1-4.3) and Definition 4.3 and 4.8, the set of all feasible control inputs is defined as
U :=
uk ∈U|xk ∈ X, wk ∈W, ∀k≥0
Remark 4.3 It should be noted that feasible input set U differs from the admissible input U, such that, in general, U ⊆U, ∀xk ∈ X, i.e., the feasible input set U is the subset of admissible input set U such that xk ∈ X, ∀k ≥ 0 and the control input satisfies the given input constraint for any disturbance within the bounded set, W, for closed-loop system xk+1 =f(xk, uk)⊕W.
Definition 4.10 (Feasible Initial condition set) The set of all initial statesx0, for which a feasible control policy exists is defined as Feasible Initial condition set, i.e.,
X0 :=
x0 ∈ X ⊂Rn
U 6=∅ . (4.5)
For a system perturbed with some persistent arbitrary but bounded disturbance as given in eqn.(4.1), it is obvious that the system states cannot be steered to the origin. However, the states can only be steered to a target set in the neighbourhood of the origin. The problem is finding a control law such that the system states reach a target set in a finite number of steps, despite the disturbances on the state, is fundamentally a problem of finding the robust controllable sets.
Definition 4.11 (Robust Controllable set) [90] The i-step robust controllable set Qi(Ω,T) is the largest set of states in Ω for which there exists an admissible
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”time-varying” state feedback control law such that an arbitrary terminal setT ⊂Rn is reached in ”exactly” i-steps, while keeping the evolution of the system states inside Ω for the first i-1 steps, for all allowable disturbances, i.e.,
Qi(Ω,T),
x0 ∈ X0 ⊂Rn| ∃{uk :=κk(xk)∈ U}i−10 :{xk ∈Ω}i−10 , xi ∈ T, ∀{wk∈W}i−10
(4.6)
It is of interest to find the maximum possible set of initial states for which a time- varying feedback control law will exist, such that the closed-loop system states reach a target set for all admissible disturbance. However, the above definition may give a more conservative problem of finding set of states if an open-loopsequence of control inputs drive the system states to the target set, irrespective of the disturbance sequence, given as
Qoli (Ω,T),
x0 ∈ X0 ⊂Rn| ∃{uk∈ U}i−10 : {xk ∈Ω}i−10 , xi ∈ T, ∀{wk ∈W}i−10
. (4.7) The inclusion of the constraint that the control input be dependent on the state as well as time makes the fundamental difference between the feedback and open- loop robust MPC design (c.f. Section 1.3.2). Interestingly, in the absence of any disturbance Qi(Ω,T) =Qoli (Ω,T).
Remark 4.4 It should be noted that if the target setT is RCI, then a time-invariant feedback control law will ensure that the states lie insideT for all time, after the states reach T in i-steps. Moreover, by definitions (4.2) and (4.3), if a time-invariant control law is chosen thenT will be a RPI set, such that the closed-loop states enter T in exactly i-steps.
Definition 4.12 (Robust Stabilisable Set) [90] The set Sk(Ω,T) is robust sta- bilisable set contained inΩfor the perturbed systemxk+1 =f(xk, κk(xk))+wk, ∀wk ∈ W iff T is a RCI subset of Ω and Sk(Ω,T) contains all the states in Ω for which there exists an admissible time-varying feedback control law κk(xk) ∈ U which will
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drive the system states toT inisteps or less, while keeping the evolution of the state inside Ω for all allowable disturbance sequences, i.e.,
Sk(Ω,T),{x0 ∈ X0 ⊂Rn| ∃uk:=κk(xk)∈ U, xk ∈Ω :x¯k∈ T ⊂Ω, ∀¯k ≥i}(4.8) The difference between robust stabilisable set and robust controllable set arise w.r.t.
the nature of the target set. In many control problems, the target set is either a robust control invariant set or a robust positively invariant set for a Lyapunov-stable closed-loop system. If the initial state is contained inside a robust stabilisable set then a control law guaranteeing that the target set T is reached by the closed-loop system states in a finite number of steps can be designed. Once the states are inside the target set one can switch to a time-invariant Lyapunov-stable controller, as used in dual-mode MPC design [113].
Definition 4.13 (Maximal Robust Stabilisable Set) [90] The setS∞(Ω,T)is the maximal robust stabilisable set contained inΩfor the systemxk+1 =f(xk, κk(xk))+
wk iff S∞(Ω,T) is the union of all i-step robust stabilisable sets contained in Ω.
Remark 4.5 From the above discussions, it can be understood that the largest pos- sible region of attraction to the target set is equal to the maximal robust stabilisable set.
In general, the maximal robust stabilisable setS∞(Ω,T) is not equal to the maximal control invariant setC∞(Ω), even for linear systems. S∞(Ω,T)⊆ C∞(Ω) for all RCI set T. The set C∞(Ω)\ S∞(Ω,T) includes all the initial states from which it is not possible to robustly steer the system states to the stabilisable region S∞(Ω,T) (and hence to T) [90].
Definition 4.14 (Optimally Feasible set of States) The feasible set of states of the optimal control problem, with the optimal control law uk = κk(xk) ∈ U, is defined as
XF :=
xk ∈ S∞(Ω,T)|∃κk :S∞(·)→ U .
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Remark 4.6 It should be noted that the set of all feasible states may differ from the maximal robust stabilisable set, S∞. In general, xk ∈ XF ⊆ S∞, ∀k ≥0.
With the above definitions on the feasible states and control inputs, the propo- sition given below follows immediately.
Proposition 4.2 In the design of robust feedback MPC for the discrete-time dy- namic system given in eqns. (4.1)-(4.3), the following statements are equivalent;
1. when the system states lie in the feasible set as xk ∈ XF, then there exists a control law(uk :=κk(xk)), such that uk∈ U, for the given state measurement.
2. when the control input uk for the given state measurement is feasible, uk ∈ U, then it implies that the states lie in the feasible set, xk ∈ XF.
With these set theoretic properties in hand, the concepts of Lyapunov theory is briefly revised to subsequently make the proper relation of the set theoretic ideas in the light of Lyapunov function to establish the stability and robustness of the closed-loop system using mixed H2/H∞ MPC designs.