This chapter introduced the general class of systems and robust controllers that will be employed to tackle some of the challenges related to controlling uncertain and changing constrained linear system. Well known results related to tube-based MPC controllers were presented and a minor modification was proposed to enlarge the RoA of the tube MPC variant that does not re-optimize the nominal trajectories, described in [1].
It is important to emphasize that although Section 2.2 introduces constrained LTI systems as the object of study, the tools presented here can be exploited
to control constrained LTV systems. The key feature that allows this is to observe the changing/uncertainty of the system as a disturbance affecting an, otherwise, invariant system. The following chapters address the advantages and drawbacks of this rationale in different arrangements.
Chapter 3
Tube-Based Adaptive Model
Predictive Control with
Persistence of Excitation
3.1
Introduction
A key element of any MPC controller is, certainly, the mathematical model used to make the necessary predictions. As discussed in Chapter 2.1 standard MPC controllers have a certain degree of robustness against prediction errors, however uncertainties in the model can lead to large mismatches and therefore have a significant impact on the overall performance of the controller [39,40]. Moreover, several MPC implementations, including those presented in Chapter 2.1, rely on the computation of invariant sets. The latter are highly model dependent, therefore model mismatch can not only decrease performance but also result in constraint violation or even unstable behaviour of the closed-loop.
It seems then that the necessity of a precise prediction model may be one of the main drawbacks of any MPC implementation due to the challenges related to its acquisition. First-principle approaches, for example, may result in models that are too complex for controller design [41] or hinge on a-priori simplifications that neglect important input-output interactions. Even if an adequate model is obtained through such techniques several obstacles remain on the path to a successful MPC implementation: large non-linearities, expected degradation due to normal operation (resulting in changes to the values of the model parameters), expected structural changes due to operation conditions (such as payload changes) and the explicit realization of the plant (i.e. the uncertainties related to its manufacture [42]). On the other hand, the cost of
system identification experiments may be prohibitively large [43], particularly if they ought to be regularly repeated due to degradation or plant changes.
In order to improve the performance of MPC controllers in the context of uncertain and changing systems, it is necessary to obtain new and more accurate descriptions of the current condition of the plant throughout its life-time. Moreover, in order to avoid expensive experiments and account for continually changing systems, it is necessary to obtain these models on-line. To do so, a form of system identification ought to be coupled with the controller. This combination is commonly known as the dual control problem [44], given that it confronts two incompatible objectives: the controller aims to regulate the system towards a desired optimal operation point while the identifier requires the system to be constantly disturbed, in order to accurately estimate it [3]. Moreover, it is necessary not only to identify a better model on-line, but also use this new and better description of the system to provide more accurate predictions for the MPC optimization. Changing the prediction model of an MPC controller while maintaining its key control properties, however, is not trivial due to the high model-dependency of most of the controller’s design parameters. This adds a second layer of complexity to the challenges inherent to the dual control problem.
The techniques devised to solve these problems, within the MPC context, are usually grouped under the label of Adaptive MPC (AMPC) [45], however this concept has been occasionally misused. For example, the techniques presented in [46–48] are referred to as AMPC approaches, however gain-scheduling or time-varying MPC might be more appropriate. Indeed, a look-up table is constructed in [46–48] by successive linearisation of a non-linear model around different operating points. At each time instant, the model used for predictions is chosen depending on the current state, however no stability guarantees are provided. Similar misconception is found in [49], where the prediction model is computed as a fuzzy weighted combination of an ensemble of linear models that represent the plant throughout the entire state space. A more rigorous approach is found in [50, 51], where the prediction model is properly defined by a linear time varying model, however this is obtained via constant linearisation of a known, thus not uncertain, non-linear model.
There are, generally, three properties that can be used to categorize AMPC algorithms:
• Are there any closed-loop stability and/or constraint satisfaction guaran- tees?
• Is the system identification (parameter estimation) algorithm guaranteed to converge to an accurate description of the true plant?
• Is the new model effectively used to update the MPC prediction model? The latter is of paramount importance, since it is the characteristic that allows to take advantage of the information provided by the system identification algorithm. Adaptation of the prediction model is present in almost all current AMPC algorithms, however the updating is performed in different ways owing to the type of uncertainty considered and the MPC technique employed. In general, most approaches proposed to date either fail to guarantee closed-loop stability, constraint satisfaction and/or estimation convergence guarantees. Nonetheless, many provide good solutions for certain aspects of the problem, and thus are important to contextualize the solution proposed here.
3.1.1
AMPC without estimation guarantees
There exist many application driven solutions to the AMPC problem that avoid almost any type of rigorous analysis [52–57]. These usually resort to suitable assumptions on system behaviour (such as open-loop stability) that fit the purpose at hand, but greatly reduce their applicability. In [53], for example, a fuzzy supervisor overviews the closed-loop behaviour and, based on some arbitrary performance criteria which include a numeric evaluation of stability, adapts the controller by modifying some of its design parameters (weights and terminal conditions).
A simple, yet effective way of formally addressing a possible mismatch between plant and model is to characterize it with respect to a predefined model structure, bound it, and treat it as a disturbance affecting an, otherwise, invariant plant [40, 58–60]. The model structure and the particular selection of its nominal parameters are not trivial to obtain but may be available from first-principle approaches and previous identification experiments. Furthermore, quantifying the mismatch may prove to be a challenging task, however a priori knowledge about the plant and its operation program are usually enough to properly estimate the expected mismatch (for example the stiffness variation of certain mechanical parts due to increased operational temperature, or the expected payload changes on unmanned vehicles). If the mismatch has been ac- curately estimated, robust approaches such as tube-based ones can be employed to regulate a nominal representation of the plant subject to the prediction error induced by model mismatch. Although this approach addresses the uncer-
tainty, if no form of system estimation is coupled with it, the mismatch itself remains un-addressed. This implies that only robust stabilizability and robust constraint satisfaction can be achieved, albeit true external disturbances may be completely absent.
A number of recent robust-based approaches have focused on the control guarantees when implementing AMPC algorithms. Uncertain continuous time state space models subject to state and input constraints are considered in [59]. The adaptive estimation algorithm provides not only an estimate of the model parameters, but also an estimate of the error bound. This bound is included in a comparison model robust MPC [61] in order to reduce uncertainty and guarantee robust stability and constraint satisfaction via standard Lyapunov arguments. The prediction model is updated by the estimates only when the information provided by the data reduces the uncertainty on the parameters. This is quantified by checking whether the smallest eigenvalue of the inverse of the information matrix is larger than at the previous step. However, this is not guaranteed to happen at any time instant.
A similar approach is developed in [60] but for a class of non-linear contin- uous time systems with a parametric affine type of uncertainty. In this case the estimation algorithm guarantees unbiased estimates [3] alongside with a non-increasing parameter error bound, however, as in [59], the latter is not guaranteed to decrease at each time instant. A min-max robust MPC imple- mentation is proposed to account for the uncertainty arising from the model mismatch. The recursive estimation algorithm is included in the MPC predic- tions in order to account for future estimation and reduce conservativeness. The latter, however, may lead to constraint violation because predictions may be far off the true plant behaviour. This becomes evident in that the terminal constraint set and associated terminal cost have to be computed accounting for all possible values of the true estimation error. Overall, the min-max opti- mization is considered a computationally intractable problem, and therefore replaced by a Lipschitz-based worst-case approach similar to the robust MPC technique in [23]. This is extended to account for external disturbances in [39] and to discrete time systems in [62].
The algorithm proposed in [63] also resorts to a robust approach similar to that in [23], but the algorithm is only applicable for open-loop stable plants. At initialization a set of all models that may represent the plant is supposed to be known in polytopic form. Every time step a set membership identification algorithm updates this set in a recursive inclusion fashion, and the prediction
model used for the MPC is selected as the centre of the largest ball contained in this set. Recursive feasibility of the optimisation is secured by an additional group of constraints designed to ensure that the output of any model inside the current set satisfies the output constraints. For MIMO systems, these additional constraints are first introduced as a set of linear programming problems (similar to a min-max optimization problem) but then transformed into a set of auxiliary decision variables. This, although straightforward, increases the computational complexity of the problem.
It should be clear that predictions cannot be considered as reliable data for the parameter estimates computation, since they are not real measurements. In [64], however, a-priori knowledge on the rate of change of the uncertainty and the error bound provided by the estimator are used to predict a set that contains the uncertainty throughout the prediction horizon, given the current conditions. This time-varying error bound is fed to the MPC controller, unlike in [39, 60] where the bound was fixed, and therefore uncertainty is decreased. A min-max MPC algorithm is shown to guarantee ultimate boundedness of the closed-loop under several assumptions that include the existence of an invariant set to be used as terminal constraint. However, constraints may be violated, and again, the parameter error is not guaranteed to decrease at each time instant.
In all of the above, the obtained estimates are not necessarily convergent to the true plant parameters, because proper excitation of the closed-loop system is assumed rather than guaranteed. A similar set-up leads to an analogous outcome in [65, 66], where constrained polytopic linear difference inclusion (pLDI) systems are considered. In this case the pLDI structure is exploited in order to address the uncertainty in a more structured way when compared to [39,60]. Assuming a convex and bounded set of possible parameters, arguments from the robust controller proposed in [26] are employed to compute parameter dependent terminal conditions. A considerable advantage with respect to [60, 64] is that constraints are satisfied even if the parameters are slowly changing within their initially assumed bound. A drawback of embedding a strong structure in the controller’s design is that, to maintain feasibility of the optimization, newly estimated parameters can only be included at the end of the prediction, moving forward one step at each time instant. Furthermore, there is no discussion about the associated estimator, and it is only assumed that a convergent one exists.
cost function is enhanced in order to push for convergent estimates by adding a term that depends on the covariance matrix of the estimates. It is expected, rather than guaranteed, that this will promote input sequences that reduce the uncertainty on the estimates, hence reducing the size of the covariance and ultimately yielding a standard MPC cost once the true model is known. Closed-loop stability and constraint satisfaction are guaranteed by means of a robust invariant set for the pLDI structure (similar to [65]), and a Lyapunov type constraint on the first prediction step. The modification of the cost results in a non-convex problem which is addressed by either the inclusion of relaxation variables, or the separation of the problem into two optimizations. The second approach is discussed in more detail in [67], where the additional term in the cost function penalizes the deviation of the input trajectory from an optimal probing sequence. The latter is obtained through a preceding step of non-convex optimization that maximizes the minimum eigenvalue of the inverse of the information matrix as in [59].
A noticeably different architecture, yet also lacking a proper convergence guarantee for the estimator, is employed in [40]. The core idea is to decouple the control and performance objectives by maintaining two models of the plant. A nominal model is employed to characterize the parametric uncertainty and design a robustly stabilizing tube MPC controller, while a second model, initialized as the nominal model, is constantly updated by an estimator. Both are employed by the controller to make predictions; those from the first model are used to ensure robust constraint satisfaction, while those from the second are used to compute the cost. In this way robust stability is maintained even if the estimates render the initial tube controller infeasible, but performance is possibly improved by using a more accurate model for evaluating the cost.
A similar approach is presented in [68], where nonlinear systems are studied. A machine-learning approach is used to obtain, off-line, a nominal model of the plant given some previous data. This is accompanied by an estimation of a Hölder constant, which allows to compute a bound on the prediction error associated to this nominal model. This bound depends on the length of the prediction, and is used to properly tighten the constraint sets in an otherwise standard MPC optimization, in order to guarantee constraint satisfaction and input-to-state stability with respect to the prediction error. This is in similar fashion to the robust MPC approach depicted in [23], but without parametrizing the control action. It is then proposed to use the closed-loop data to continuously obtain more accurate models, however the different MPC
elements used to guarantee recursive feasibility of the optimization have already been computed for the model obtained off-line. This results in that newly obtained models can only be used to evaluate the cost, while the off-line obtained one continues to be used to guarantee constraint satisfaction.
In a similar fashion, the controller proposed in [69] attempts to decouple the goals of control and system identification in order to provide a solution to the dual control problem for uncertain LTI systems. The main drawback is that both tasks are executed separately, employing the whole input capabilities, thus open-loop stability is a required assumption. A technique known as zone- tracking MPC [70, 71] is employed to steer the uncertain system towards an invariant set for identification, or more simply a set that is robustly invariant against model mismatch and persistently exciting inputs. While inside, a previously defined persistently exciting sequence can be implemented, for as long as necessary, in order to accurately estimate the model parameters. Attractivity of the target set for identification is guaranteed for the nominal model of the plant; although open-loop stability guarantees convergence of the true plant to the set, only nominal trajectories are shown to be constraint admissible, thus the mismatch may result in constraint violation. This issue is dealt with in [72], by employing a robust MPC formulation, but several other drawbacks remain, such as the requirement for open-loop stability.
The concept of homothetic tube MPC is coupled with a set-membership identification algorithm in [73] to produce an adaptive MPC scheme that guarantees constraint satisfaction and practical stability of the control target. The set-membership identification approach, guarantees that the estimated set that contains the true parameters is non-increasing in size. This allows the homothetic tube MPC, which is also a set based MPC controller, to guarantee recursive feasibility of the optimization problem (and hence recursive constraint satisfaction). As opposed to standard tube MPC, the homothetic approach reduces conservativeness as it updates the tightening on the constraint sets at each time instant and throughout the prediction horizon. This update takes into account the new information provided by the estimator, and also the fact that the parametric uncertainty decreases in size as the state and input trajectories approach the origin. Nevertheless, practical stability is achieved, by modifying the standard MPC optimization problem into a min-max problem, and the set-membership approach only guarantees a non-increasing set that contains the true parameters. Furthermore, this implies that the proposed approach is not applicable for uncertain time-varying plants.
3.1.2
AMPC with estimation guarantees
The main motivation for designing and implementing an AMPC algorithm is to address uncertainties in the model and changes in the plant while the latter is in operation, thereby improving the performance of a predictive controller without incurring expensive or prohibitively long down times. Therefore, it is necessary to guarantee that the algorithm set forth to obtain the estimates will effectively yield a more accurate representation of the plant (provided it is allowed by the model structure selected [3]). Furthermore, particularly relevant to robust AMPC approaches is to compute an a-priori bound on the prediction error, ultimately meaning that a valid estimate ought to lie inside a bounded set of models. Guaranteeing convergence, uniqueness and boundedness of the estimates is not a trivial task, especially since the on-line nature of the process implies that closed-loop system identification is to be performed. Indeed, it is easily shown that many parameter estimation algorithms may result in biased