1.3 Robust Model Predictive Control
1.3.2 Open-loop & Feedback robust MPC
For almost the past four decades, the control problem of driving the states to a target set (preferably in the neighbourhood of the origin), for a constrained system
1.3. Robust Model Predictive Control 16
subject to persistent disturbance, was solved by minimising some worst case cost [161, 49, 15, 16, 64]. In such a problem, also known asminimax feedback control, the major issue had been how the system states’ evolution could be kept inside the target set. This has been addressed in [112]. They considered the target set to be robustly invariant. Once the states are inside the target set, the control input is determined by a pre-computed control law which ensures that the state trajectories never leaves the target set. In [15],[64], [23] and [112], the solution to this problem was provided on the basis of set-theory and set invariance property. A detailed account of the set invariance theory and its role in robust control problem can be found in [25, 26, 90].
Although satisfying the terminal conditions (c.f. Section 1.2.2 a-c) are necessary for robust and stable control of an uncertain system (eqns. (1.22) and (1.23)), the computational cost associated with it for obtaining the solution on-line (even for a finite-horizon approximation of the infinite-horizon problem) might be too expensive.
In such a case, it is necessary to consider all possible realizations ofxu,w(·;xk)∈F in the optimal control problem and it has to be ensured that each realization satisfies the constraints and the terminal conditions [113]. To address this computational issue, as an alternative approach, an explicit expression of control law using multi- parametric approach was proposed [8, 9, 10]. Here, a piecewise affine expression of the control law is computed off-line. There are also methods to address this robustness issue which rely on the min-max optimisation methods of the predicted performance [101].
As an appreciation of the strength of feedback in the light of robustness for MPC design, Mayne [109] proposed the concept of feedback min-max MPC. Generally in MPC design, even min-max MPC, a single control input sequence is computed with an aim to minimise the worst case cost and only the first control value is applied to the plant and the same control action is retained until the next new measurement comes. This is calledopen-loop min-max MPC. On the other hand, in feedback min- max MPC, the notion of feedback present in the receding-horizon implementation of the control, is used [139]. One such feedback MPC methodology is used in this
1.3. Robust Model Predictive Control 17
thesis but in an entirely different setting such that the problem is posed and solved using Nash game approach, which are detailed the subsequent sections.
(a) Open-loop min-max MPC
Consider a nominal system, whose stable closed-loop is given by xk+1 = f(xk, uk).
If there exists a positive invariant set and xk ∈ XN ⊂ Rn, then the condition xk+1 ∈ XN−1 ⊂ XN also prevails. However, this property is no longer valid in presence of some uncertainty (eqn. (1.22)). Nevertheless, this property can be recovered by considering all the possible realizations of the xk ∈F that satisfy the state and/or control constraints and terminal conditions, for all admissible set of disturbances in open-loop, Wol, for computing the optimal control input sequences.
The problem to be solved in an open-loop robust MPC is given below;
Problem 1.2 Solve min
uN
k
max
{wk∈Wol}N−10
F(xN|k) +
N−1X
i=0
L(xi|k, ui|k)
(1.24) subject to,
xi+1|k =f(xi|k, ui|k, wi|k), x0|k=xk, ∀k ≥0 (1.25) xi|k∈X, ui|k ∈U, i= 0, ..., N −1 (1.26)
xN|k∈ Xf (1.27)
The decision variable uNk is
uNk ,[uT0|k, uT1|k, ..., uTN−1|k]T (1.28) where F(·) and L(·) represents the terminal and stage costs, respectively. Thus, in open-loop min-max MPC a single control input sequence is used to minimise the worst case cost. Let, for alli≥0,Xol
i ⊂ Xi is the set of states that can be steered to Xf, in a finite number of steps ior less for all admissible open-loop control sequence uNk. If there exists a finite N > 0, such that xN|k ∈ Xf, then Xol
N is the domain of attraction of the open-loop robust MPC for the set of admissible control sequence
1.3. Robust Model Predictive Control 18
uNk ∈Uol
N 6= 0.
However, in general, the open-loop formulation is too conservative and may often under-estimate the set of feasible trajectories [93]. This is due to the reason, that a single control sequence is computed such that for all allowable disturbance sequences (wk ∈ Wol) the constraints are satisfied. Moreover, the open-loop min-max MPC assumes that the control input sequence (uNk) computed at timek will be applied in N steps in the future, without considering the fact that the states will be measured and a new control sequence will be recomputed at each of the subsequent time steps.
(b) Feedback robust MPC
In an open-loop min-max MPC there are always chances that the trajectories satis- fying xk∈F may diverge, as the uncertainty may spread over the horizon, causing the set Xol
N to be small or even empty for a reasonable N > 0. On the other hand, incorporating the notion of feedback available in MPC in the form of feed- back control law (κk(xk) : X 7→ U, ∀k ≥ 0), instead of a sequence of control actions as in open-loop control problem, prevents the trajectories from diverging once xk+1 = f(xk, κk(·)) +wk is stable. For this reason, the idea of feedback MPC was proposed by Mayne [109] and later such idea was used in [96, 139, 111].
The problem of feedback robust MPC is given as follows [111]
Problem 1.3 Solve min
πNk max
{wi|k∈Wf b}N−10
F(xN|k) + XN
i=0
L(xi|k, ui|k)
(1.29) subject to,
xi+1|k=f(xi|k, πkN, wi|k), x0|k=xk, ∀k ≥0 (1.30)
xi|k∈X, i= 0, ..., N −1 (1.31)
{u0|k, κi|k} ∈U, i= 1, ..., N −1 (1.32)
xN|k∈ Xf (1.33)
1.3. Robust Model Predictive Control 19
The decision variable πNk is
πkN := [u0|k, κ1|k(x1|k), ..., κN−1|k(xN−1|k)] (1.34) where the sequence of control actions in open-loop robust MPC (1.28) is replaced by a control policy (1.34).
Note that κi|k(·) for every i ∈ {1,2, . . . , N −1} are control laws but u0|k is a control action, since there is only one deterministic initial state if all the states are completely measurable. Let Wf b be the set of admissible disturbance if the control policyπkN is used and ΠN be the set of all admissible control policies (πNk ) of lengthN satisfying the constraints and terminal conditions. Then, for alli≥0, letXf bi denote the set of states that can be steered to the robust invariant setXf, in a finite number of steps i or less, by the admissible control policy πNk such that Πi 6= 0; Xf b
i ⊂ Xi. Thus for someN >0 ifxN ∈ Xf, andXf b
N is positive invariant, for the given system, then Xf b
N is the domain of attraction.
As the open-loop min-max MPC is more conservative than the feedback robust MPC, the domain of attraction of the later is often much larger than that of the former, i.e.,
Xol
N ⊂⊂Xf b
N
However, such a min-max feedback MPC scheme is difficult to implement due to its prohibitive computational cost. Moreover, it may become intractable as the hori- zon length is increased. To address this issue, inclusion of an additional robustness constraint is proposed in [39].