Robust Model Predictive Control and Techniques to Enforce Stability

Below we briefly review some of the popular techniques used in the literature to

“enforce” closed loop stability. A robust control system can account for bounded disturbances while still ensuring that state constraints are met.

Morari and co-workers were responsible for the early efforts into adapting the robust control framework to include MPC schemes. Works such as Campo and Morari (1987), Kothare, Balakrishnan and Morari (1996) and Bemporad and Morari (1999) provided the basis for robust model predictive control by formulating the cost functions as Lyapunov equations, resulting in the nominal stability of the closed control loop. Lee, Morari and Garcia (1994) is also a very relevant paper which helped establish state-space models as the standard for modern MPC formulations. Finally, MPC surveys such as Garcia, Prett and Morari (1989) and Morari and Lee (1999) provided interesting and useful overviews on robust MPC theory for beginners as well as experienced researchers.

The seminal idea that prompted the establishment of the robust MPC framework is the set of “Terminal Constraint” techniques, an approach firstly proposed in Kwon and Pearson (1977). According to Bemporad and Morari (1999), this set of techniques can be divided into two main classes. In the first class, the objective function is defined in such a way that it corresponds to a Lyapunov function. The second explicitly requires that the difference between current state and reference shrinks in some norm through the prediction horizon.

The main drawback of using terminal constraints is that the control effort required to steer the state to the reference can be large, especially if a short control horizon is used, and therefore feasibility is a critical issue. Feasibility is limited to the “domain of attraction” of the closed-loop (MPC+plant) is defined as the set of initial states that can be steered to the reference values. This is a problem because the range of operating points where a plant is expected to operate may be considerably larger than the set of initial states which steerable to the reference in an arbitrary number of steps. Also, the speed of control actions can be negatively affected because of the artificial terminal constraint. A variation of the terminal constraint idea has been proposed where only the unstable modes

81 are forced to zero at the end of the horizon (Rawlings and Muske, 1993). This mitigates some of the mentioned problems.

Let us briefly discuss other important contributions to the robust MPC field.

Nominally stable MPC algorithms that use terminal constraints include: Infinite Output Prediction Horizon (Keerthi and Gilbert, 1988; Rawlings and Muske, 1993; Zheng and Morari 1995); the terminal Weighting Matrix (Kwon et al. 1983; Kwon and Byun 1989);

Invariant Terminal Set (Scokaert and Rawlings 1996); Contraction Constraint (Polak and Yang 1993a,b; Zheng 1995).

Mhaskar, El-Farra and Christofides (2005) presented a Lyapunov-based model predictive control framework for switched nonlinear systems (models that switch at prescribed times). Mhaskar, El-Farra and Christofides (2006) consider the problem of the stabilisation of nonlinear systems subject to state and control constraints. An auxiliary Lyapunov-based analytical bounded control design is proposed to characterise a “stability region” of the MPC and also provide a feasible initial guess to the optimisation problem.

It is also worth citing Heidarinejad, Liu and Christofides (2012) which proposes an Economic MPC (EMPC) of nonlinear process systems using Lyapunov techniques, and Liu, de la Peña and Christofides (2009) and Christofides et al. (2013) which introduce the concept of Distributed MPC, a robust MPC scheme which establishes communication between several different MPC controllers in order to achieve better closed-loop control performance. Specifically, each of the distributed algorithms comprehends both a local controller, who optimises a local (non-cooperative) cost function and also a global controller which handles a global (cooperative) cost function that refers to other distributed algorithms.

Allgöwer and co-workers contributed to several widely cited works on robust MPC theory. Among them we could highlight Chen and Allgower (1998), Allgöwer et al. (1999) and Findeisen and Allgöwer (2002), all of which helped establish the field of Nonlinear Model Predictive Control (NMPC) taking advantage of better nonlinear optimisation algorithms and increased computational power available in the late 1990’s and early 2000’s. NMPC is making use of nonlinear state-space models do better predict the process outputs and provide more efficient control actions, but it also results in a control problem for which is much harder to find the globally optimal solution.

Odloak and co-workers contributed to several works on Infinite horizon MPC (IHMPC), including González, Perez and Odloak (2009) and Rodrigues and Odloak

82 (2003a), which expanded IHMPC to integrating processes, and González and Odloak (2009), which introduced a Zone Constrained IHMPC (see Section 2.2.2). We may define IHMPC as an inherently robust MPC framework for constrained linear systems, which makes use of an infinite prediction horizon by replacing the infinite horizon objective by a finite one after defining a penalty weight matrix (terminal cost) at the end of the input horizon. The terminal weight is obtained from the solution of a discrete-time Lyapunov equation. Stability is guaranteed as long as the related optimisation problem is feasible.

Also, features were added that enabled the IHMPC to handle common industrial problems such as zone constrained MPC and integrating processes, as well adapting IHMPC for use with models with polytopic uncertainty (Rodrigues and Odloak, 2003b), and providing robust integration with real-time optimisation packages (Alvarez et al., 2009).

Although terminal constraint techniques are very popular and effective for guaranteeing closed-loop MPC stability, some alternative robust MPC schemes were proposed. Some of them are outlined in Table 1.

Table 1 – Alternative robust MPC schemes.

Min-Max MPC (Scokaert and Mayne, 1998)

In this formulation, the optimisation is performed on all possible evolutions of the disturbance. This is the optimal solution to linear robust control problems; however, it carries a high computational cost.

Constraint Tightening MPC (Richards and How, 2006)

Here the state constraints are enlarged by a given margin so that a trajectory can be guaranteed to be found in any evolution of disturbance.

Tube MPC (Langson et al. 2004)

This uses an independent nominal model of the system and uses a feedback controller to ensure the actual state converges to the nominal state. The amount of separation required from the state constraints is determined by the robust positively invariant (RPI) set, which is the set of all possible state deviations that may be introduced by a disturbance with the feedback controller.

Multi-Stage MPC (Lucia et al. 2013)

This uses a scenario-tree formulation by approximating the uncertainty space with a set of samples, and the approach is non-conservative because it takes into account that the measurement information is available at every time stages in the prediction and the decisions at every stage can be different and can act as recourse to counteract the effects of uncertainties. The drawback of the approach, however, is that the size of the problem grows exponentially with the number of uncertainties and the prediction horizon.

83 2.2.2 Zone Constrained Model Predictive Control

Zone constrained MPC was briefly cited in the introductory Section and, being a key element of this project, it shall now be described in further detail. Zone constrained MPC is not a unique class of MPC algorithms, but rather an alternative way of defining the control objectives which can be applied to any MPC scheme.

In the ‘perfect’ model predictive control framework the goal is the complete rejection of the disturbances, and thus it is required that the output variables return to and remain in the original state before the end of the prediction horizon. But it does not matter how advanced or robust a controller may be if the processes do not possess an equal or superior number of unsaturated manipulated variables compared to the number controlled variables, a solution for the perfect control problem does not exist.

So in most control applications, these restrictions apply and thus perfect control is not attainable, but even so, all controlled variables need to be kept within certain limits.

This is the reason why most commercial MPC controllers in the chemical industry operate with a variation of partial control often denominated “zone control” or “zone constraints”, in which every controlled variable has maximum and minimum desired values, so that the control problem is not to keep each one at a fixed set-point but instead to keep them all inside the zones bounded by their maximum and minimum values.

However, the zone constrained MPC will not be able to keep all controlled variables within their desired control zones all of the time due to the lack of degrees of freedom, and then some restrictions are eventually violated. This may happen because there may be disturbances acting in the process, or the zones that were defined are too narrow, or perhaps the inputs to the process are already saturated. The zone control restrictions to the MPC problem are sometimes called “soft constraints” since they can be eventually violated. Likewise, every manipulated variable also has its required maximum and minimum values. The MPC controller cannot violate these restrictions; hence they are called a set of “hard constraints”. These manipulated variable restrictions are associated with physical constraints. For example, a control valve cannot have percentage opening outside the 0-100% range, or the operating temperature of a reactor must obey its metallurgical limit, and a plant feed flow cannot be negative and so on. On the other hand, the soft constraints are usually related to product or process specifications and are defined by process dynamics and thus cannot be set to a specific value forthwith. Please note that classical set-point MPC is just a special case of zone control MPC, in which

84 output maximum and minimum limits are equal. Zone control MPC also may encompass any kind economic or robust MPC, being, in reality, the most generic MPC definition.

In fact, virtually all MPC packages commercialised globally are a combination of zone constrained MPC and economic MPC, and some of them also claim to have robust performance. Among these packages, we can mention Honeywell™ MPC, Shell-Yokogawa Exa-SMOC™, Emerson DeltaV™ Predict and AspenTech DMCplus™ as offering simultaneously both process (zone) control and economic optimisation, in a combination that has become a standard feature for industrial control applications to perform. Schemes with guaranteed stability are not as popular in the industry since they result in unnecessarily slow control actions (as stated by Bemporad and Morari (1999), Robust MPC control actions may be excessively conservative).

Whereas most recent academic research has been focusing on robust MPC, comparatively fewer contributions have been made for optimising zone constrained MPC.

Some examples of research concerning zone control are found in González and Odloak (2009), Grosman et al. (2010), Luo et al. (2012) and Zhang et al. (2011). There is also a more sizeable bibliography of works covering the integration of economic optimisation and MPC control, such as Porfírio and Odloak (2011), Gouvêa and Odloak (1998) and Adetola and Guay (2010).

In zone control each controlled variable has a minimum and maximum desired variable but some of these constraints may have more importance than others and, for this reason, when defining the MPC control problem it is common practice to assign each controlled variable a weight value, which establishes the relative priority each bound will have in the solution. For example, constraints to the process, the equipment and environmental safety normally have precedence over those concerning product specifications. Henceforth these weight values that relate the comparative importance of each controlled variable will be denominated _',PNNLH and _{',)K LH}, meaning the weights respectively for the upper (maximum value) and lower (minimum value) bounds of controlled variable w_', where ¹ = 1, … , M_Q, and M_Q is the number of controlled variables.

Let us now present a quick example to further clarify Zone Constrained MPC. For instance, consider now a system controlled by an MPC controller that consists of two controlled variables, w and w , a single manipulated variable, f , and a single disturbance, ë . Also, consider that y has a higher weight in the control problem than y , so that _,PNNLH = _{,)K LH} = 2 and _,PNNLH= _{,)K LH} = 0.5, and that the zone

85 constraints bounding the controlled variables are w _,˜? = w _,˜? = 2 and w _,˜'* = w _,˜'* = −2. Let it be assumed that the models defined by Eq. 31 to Eq. 34 describe the interaction between the controlled variables, the manipulated variable and the disturbance:

Q ,P k^m =_{W p}^._{| |} Eq. 31

Qp,P k^m = ^._| Eq. 32

Q ,, k^m = _| Eq. 33

Qp,, k^m =^.B._| Eq. 34

Now consider that this system is subject to a disturbance ë k^m =^W. If no control action is taken, w will increase until it violates its upper bound while w decreases, but without leaving its desired control zone. In order to keep higher priority w within bounds, decreasing its value to the maximum limit, w _,˜? = 2 then the minimum control input necessary is f k^m = . However, this movement in the manipulated variable will further decrease w , which will thus violate its lower bound, w _,˜'* = −2. In this case, the zone constrained MPC controller will prioritise the variable whose deviation from bounds or error has the largest impact on its objective function, which means keeping w in its control zone at the expense of the less important w , since it doesn’t have enough degrees of freedom to control both. However, the controller should make the minimum movement necessary to maintain w within its control zone, minimising the error due to violating w restriction.

Let us assume that the controller takes 10 seconds to react to the ë . In this case, the system response is the one shown in figure 6. Please note that the control action is very steep in this example and an MPC controller can be tuned to provide a more smooth response. However, this has been ignored to simplify the analysis.

86 Fig. 6 – How an MPC controller handles the zone control problem.

Fig. 7 presents a diagram showing how a zone constrained MPC calculates the error (E_¬) along the whole trajectory prediction:

Fig. 7 – Error calculation for Zone Constrained MPC.

Eq. 35 defines the error related to vector of controlled variables at each instant k:

87 c_\= µ _Vh− _\· ∙ s_{de T}+ µ _\− · ∙ s _{YY T} Eq. 35

A key point here is that the definition of the control zone is related to process constraints, not to the controller; maximum and minimum values for each variable are usually provided by the operator through MPC control user interface. Safety concerns and desired product specifications are constraints to possible solutions by bounding the control zones.

2.2.3 Interfaces of industrial MPC implementations

Fig. 8 – Interfaces of an industrial MPC implementation.

Industrial implementations of MPC schemes present several interfaces and components. Fig. 8 shows the interactions between the main elements inside the blue box:

the model, which provides the output prediction; the cost function or algorithm, which evaluates the prediction according to the control goal priorities; and the optimiser or solver which searches for the optimal set of control action through the input-space.

Human operators provide limits for MVs and CVs as well as the direction and priority (weight parameters) of economic optimisation, setting the cost function parameters and restrictions for the solver to use during its search for the optimal solution. Sensors located in the process site measure properties such as temperature, pressure, level and flow rate from the relevant streams. These measurements are filtered and converted to vectors of appropriate form (states) to enable future output prediction and bias correction. This conversion is performed by a state estimator, also known as state observer, which is a

88 system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. An estimator is required when using state-space models since in most practical cases, the real state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer (Definition 2.1.1.1.2). The Kalman filter, proposed in Kalman (1960) is the most commonly employed state estimator. All layers of the MPC implementation are equally important for a successful operation.

2.2.4 Conclusions from the Model Predictive Control Review

A brief review of Model Predictive Control was presented in order to bring the reader’s knowledge of to the level required to, firstly, understand some key choices made concerning the EMOP methodology development; secondly, allowing a full understanding of the review of Integrated Process Design and MPC Methodologies in the next Section.

So the reader should now have a basic understanding of the MPC fundamentals and be aware now that it is the most popular ‘advanced control’ structure in the industry, and also the one that has been receiving more academic contributions. Particularly relevant in recent years are the contributions mostly focused in guaranteeing closed-loop robust performance, which was examined in Section 2.2.1.

Section 2.2.2 presented ‘Zone Constrained MPC’, which is the form in which MPC control problem objectives are usually defined in industrial applications. The reader should now understand that the usual approach of fixed SPs or reference values is a special case of ‘Zone Control’, itself a broader definition of MPC control goals.

Additionally, it is important to emphasise that most algorithms can be adapted as ‘Zone Constrained’ MPC, including those with guaranteed robust performance.

2.3 Review of Integrated Process Design and Model Predictive Control Methodologies

Adding MPC to the Integrated Process Design and Control framework is a very challenging task which has been only recently received the publication of several results.

The purpose of this Section is to present a comprehensive survey of such works.

89 2.3.1 Embedding MPC in Flowsheet Analysis

As discussed in the Literature Review of Methods of Integrated Control and Process Synthesis, Chapter 2, there are several interesting papers where the integrated design methodology is applied to determinate the optimal design and control structure for a given process. Most of them undertake challenging issues in the integrated design framework such as alternative procedures to evaluate controllability, uncertainties handling techniques, the inclusion of different control strategies or address a complex application. An important aspect of these works on integrated design is the type of controllers and control strategies considered. Given how necessary and widespread considered conventional feedback controllers such as PI or PID are for the control of continuous processes such as those found in the chemical industry, it is only natural that the bulk of simultaneous design and control methodologies developed for chemical processes use this kind of control structure in their analysis, and these works were covered in Section 2.1. The works that deal with advanced control structures, such as the hugely popular Model Predictive Control, are yet few. All of them, to the author’s knowledge, are discussed in this Section.

Until now, MPC schemes appear seldom in the framework’s literature because the application of advanced control strategies in the integrated design framework is limited by the complexity of the resulting optimisation problems. However, the availability of improved computational resources allowing more powerful optimisation and computing methods, together with mature Controllability Analysis tools and advanced control technologies, provide the necessary driving force to address advanced control techniques, which introduce significant improvements in the process dynamic performance, particularly in the multivariable cases. MPC has become the advanced control method of choice in the chemical industries such oil refining for mixture separation, reactors and product blending, and for this reason, its insertion in the integrated design framework is a desirable development.

Integrated design of the chemical processes and MPC controller problem consists of simultaneously determining the plant and MPC controller parameters together with a steady state working point, while the investment and operating costs are minimised.

Integrated design of the chemical processes and MPC controller problem consists of simultaneously determining the plant and MPC controller parameters together with a steady state working point, while the investment and operating costs are minimised.