MPC incorporates constraints explicitly, is easy to formulate as a constrained optimization problem, and solve with standard software. It took off in the process industry, where objectives such as the throughput or the profit dependent on it could be directly used as an objective function. Shell started using MPC in the 1960s, and Honeywell had a proof of its convergence/stability by the late 70s/early 80s in an internal report. Academic work on MPC, which began perhaps in the late 1980s showed how to reduce the computational cost of MPC in important problems, as running optimization in real time wasn't very easy in the 1990s/early 2000s when computers weren't that fast (see the works of Camacho, Morari and co-workers such as Bemporad and Borrelli).

In the process plants of today, MPC is almost exclusively used, in conjunction with system identification to capture changing process dynamics. This is needed on the one hand because of the changing quality of iron ore, or crude oil entering a steel mill or a refinery respectively, and possible on the other hand because of the relatively slow  dynamics of these processes compared to the frequency of measurement, or that of computation. Moreover, process plants have extremely powerful actuators (valves, stirrers and so forth) that make for extremely robust performance.

MPC is now moving into several areas where only fast reactive controls were used, such as in automobiles and airplanes. It also provides a framework for any constrained dynamic optimization, such as supply chains and logistics, or autonomous missions for UAVs/drones/robots. Its main disadvantage arises where the models are not known very well, because of its strong dependence on the model. However, in a world of fast improving measurement, this may not be as much of a difficulty.

LQ technique is important for several reasons:

It provides a set of analytical design procedures that facilitate

the synthesis of control systems with nice properties.

LQ control allows us to design control systems under various assumptions on the information available to the controller.

If this includes also the knowledge of the reference to be tracked, feedback as well as feedforward control laws---the so called 2 DOF(Degree OF Freedom) are jointly obtained analytically.

LQ control theory can be applied to nonlinear systems operating on small signal basis.

There exists a relationship of duality between LQ control and Minimum-Mean-Square linear prediction, filtering and smoothing. Hence, any LQ control result has a direct counterpart in the latter areas.

LQ and predictive control:

With respect to LQ control, predictive control basically adds constraints in the tracking error and control variables and

uses the receding horizon control philosophy. In this way, relatively simple 2 DOF control laws can be synthesided. Their feature is that the profile of the reference over the prediction horizon can be made part of the overall control system design and dependent on both the current plant state and the desired set point. This extra freedom can be effectively used so as to guarantee a bumpless behavior and avoid to surpass saturation bounds. These are aspects of such an importance that multistep predictive has gained wide acceptance in industrial control applications.

Predictive vs. LQ control:

There are lot of types of Predictive control. The concentration will be predictive control within the framework of LQ theory which is useful when the plant is only partially known. In fact, a predictive control law referred to as SIORHC (Stabilising I/O Receding horizon control) is singled out by addressing a dynamic LQ problem in the presence of suitable constraints on its terminal regulation/tracking error and control signals. SIORHC has the peculiarity of possessing a guaranteed stabilizing property, provided that its prediction horizon length exceeds or equals the order of the plant. This finite horizon stabilizing property makes SIORHC particularly well-suited for adaptive control wherein stabilization must be insured irrespective of the actual value of the estimated plant parameters. SIORHC, as its prediction horizon becomes larger, tightly approximates the steady-state LQ control, when the latter optimally exploits the knowledge of the future reference profile. However, thanks to the finite length of its prediction horizon, SIORHC can be more easily computable than steady-state LQ control, since it does not require the solution of an algebraic Ricati equation or a spectral factorization problem.

Optimal control generally refers to open-loop control, while MPC is closed-loop control: In both methods you use a model to predict the system behaviour. In optimal control you compute a sequence of input signals that steer this system. If you apply this input sequence to the real system, there is usually a deviation between the predicted an actual behaviour due to system-model mismatch and disturbances. You either have accept this deviation or combine optimal control with a closed-loop method. In MPC, the correction is built in, because you recompute your optimal sequence after periodically resetting the initial conditions in the MPC problem to the true state of the system (or an estimation thereof). 

It depends on your application. MPC is a closed loop control, optimal control is an open loop one. If you want to use MPC, I advise you to read this book:

Tilte: Model predictive control

Author: Camacho, Eduardo F and Alba, Carlos Bordons

Year: 2013

Publisher: Springer Science & Business Media

From my perspective it is crucial to specify the considered setting to answer your question:

First consider the following infinite-horizon optimal control problem

minu(.) J(x0,u(.)) = \int_{0}^{\infty} F(x(\tau), u(\tau))d\tau

subject to

dx/dt = f(x,u), x(0) = x0

where J(x0,u(.)) is the objective functional to be minimized. In theory it is very tempting to search for the optimal input u(.) as a feedback, i.e. as an explicit function of

u(t) = K(t, x)

Yet, the catch is that except for special cases it is very hard or even not possible to compute the optimal input as a feedback. This is due to the difficulties of solving Hamilton-Jacobi-Bellmann Equations for nonlinear systems subject to constraints.

On the other hand, we have very powerful numiercial methods to compute the optimal input fo finite-horizon problems

minu(.) J(x0,u(.)) = \int_{0}^{T} F(x(\tau), u(\tau))d\tau

subject to

dx/dt = f(x,u), x(0) = x0

as an explicit funciton of time. We may even consider input and state constraints. In this case the optimal input u(.) is an open-loop input which clearly depends on the initial condition x0, i.e.

u(t) = u(t, x0).

In MPC one relies on the tools to compute u(t, x0) fast for the curent values of x0.

As Martin Mönnigmann said previously this yields a feedback which is easier to attain than the optimal explicit feedback. Hence, whenever there is any source of plant-model mismatch, MPC has advantageous feedback properties, while (open-loop) Optimal Control does not. Furthermore, the receding horizon nature of MPC allows handling infinite horizons, while this is not easy to do in general problems of Optimal Control.

I don't understand why you say MPC is better than LQR. LQR by definition is the best control for closed loop performance. In fact, dual mode MPC usually uses LQR as the control law for the first mode (why would that be). And I also, I don't understand why people refer to LQR as "open-loop" control, like u=-Kx of LQR is open loop? Nonsense. Quite the oposite actually. Moreoveer, LQR has guaranteed stability properties, infinite gain margin and phases margin above 45, whereas MPC stability depends on the length of your prediction horizon and the properties of the system you are trying to deal with. 

I'll give you a numerical example. Try to find an LQR and MPC for this discrete plant.

yk=2.5 y{k-1} -1.25 y{k-2} + 0.5 u{k-1} + 0.7 u{k-2}

For your information its an unstable and non minimum phase plant. 

Arrange in non minimum realization state space so you can compute lqr from Matlab dlqr command directly.

A=[2.5,-1.25,0.7; 1,0,0; 0,0,0]    B=[0.5;0;1]

Choose Q weighting the output only and any R and get LQR

Q=diag([1,0,0])       R=1

U get, Kx=dlqr(A,B,Q,R)

Kx=[2.0980 -1.3245 0.7417]

Try to get that using MPC with big horizon (above 100). You should get that the hessian matrix either is close to singular or simply NaN. 

Finally, something I want you to notice. MPC finds a vector Ufuture for which the control law Ufuture=-K_xX+K_rR can be derived. If you arrange your predictions like

Yfuture=GX+HUfuture

and you minimize J=(Yref-Yfuture)'Q(Yref-Yfuture)+Ufuture' R Ufuture

then your control gains are given by, K_r=(H'QH+R)^{-1}H'Q and K_x=K_rG

and you apply only the first row of both K_x and K_r due to the receding horizon principle,

Compute your input vector (all the vector) with Ufuture=K_rYref-K_xX

where Yref is a vector of your desired reference trayectory (or value if simply want to consider it constant) and look at the value of K_x. You should see that each row has a different gain.

This is the fundamental difference of closed form solution and open form solutions.

Refer to Ogata (Discrete Time Control Systems, 2ed, chapter 8th) for the derivation of the LQR and you will notice that he uses the closed form solution where they find a single control law u=-Kx that minimizes the whole trayectory, whereas MPC says, I can have all the movements I want. This is desirable if you want to track trayectories but I strongly suggest you use dual mode so you can guarantee stabilty with the first mode.

Regards.

Dear Oscar,

Indeed you are correct that under certain assumptions (linear dynamics, quadratic positive definite cost, no constraints) the LQR delivers the optimal input in feedback form. Yet, what you somehow gloss over is that the LQR is not applicable whenever these assumptions are not met. So

- whenever one considers nonlinear dynamics and the initial condition lies rather far from the desired equilibrium

- whenever you consider input or state constraints

- or whenever the performance index is not quadratic

then the LQR is not directly or not all applicable. However, MPC at large (i.e. modulo technicalities) is applicable in these situations. Especially, in case of nonlinear dynamics one may rely on very powerful open source tools to solve the OCP approximately in real-time.

This said, it is clear that also MPC is subject to a several strong assumptions (descent model, sufficient computational resources, ...). Moreover in a discrete-time setting the receding-horizon MPC solution can be regarded as an approximation of the optimal feedback, see book of Grüne and Pannek.

To sum up, to my understanding, your justified defense of the LQR only holds under certain assumptions. 

Best

Dear Timm,

Thank you for your answer. 

Indeed, you are right. I was actually thinking about modifying my comment to say, "thats ofcourse if your system is unconstrained and linear" but oh well I thought that was implied. As for MPC/NMPC I am working on it my self. And yes, the MPC solution is an approximation of the optimal feedback. However, I just wanted to recall that MPC does have its problems as well. First, your plant can;t be unstable if you want to match the optimal feedback because your hessian grows unstable. And also, formulations like PFC have some problems when dealign with non minimum phases systems. So for that matter, I recommend prestabilization (or Dual Mode) and this usually uses LQR.

One thing I'm not sure though. You said you can't apply LQR to nonlinear. Two ideas come to my mind. 

Solving the Discrete Alebraic Riccatti Equation with matrices A and B linearized around the whole trayectory (Not sure this one would work). And also, explicit solutions of LQR for nonlinear systems, so bassicaly K as a function of the state and the input as an increment to the nominal as in Gauss-Newton method. If I understand correctly, this paper discusses how they applied it together with an NMPC dual mode scheme. ("A New Dual Mode NMPC Scheme with Terminal Control Laws of Free-Parameters")

Regards,

Oscar

Hi Oscar,

interesting reply. Thanks.

I'd like to quote and comment on a few statements from your reply though:

"However, I just wanted to recall that MPC does have its problems as well."

Totally agree to this point.

"First, your plant can;t be unstable if you want to match the optimal feedback because your hessian grows unstable."

Frankly, I do not understand what you mean. As far as I know, the results on approximation properties of NMPC with respect to infinite-horizon optimal control do not require stability of the uncontrolled (i.e. open-loop) plant. See Grüne, Lars. "Approximation properties of receding horizon optimal control." Jahresbericht der Deutschen Mathematiker-Vereinigung 118.1 (2016): 3-37.

It might be that you are on the wrong track here. Or you refer to some very special MPC variant?

"And also, formulations like PFC have some problems when dealign with non minimum phases systems."

It is a kind of philosophical question whether or not PFC is a kind of MPC. Personally, I "define" MPC as receding horizon optimal control.

"Solving the Discrete Alebraic Riccatti Equation with matrices A and B linearized around the whole trajectory (Not sure this one would work)"

That one can do this is known since the 1970s (if not earlier), see the book of Anderson & Moore. Yet where is this reference trajectory coming from? How do you compute it? How would you do this if the reference trajectory is defined over an infinite horizon? Moreover, you still go back to a linear-quadratic setting. Put differently, whenever a disturbance "kicks" the system off the reference too far, it won't work any more.

"And also, explicit solutions of LQR for nonlinear systems, so bassicaly K as a function of the state and the input as an increment to the nominal as in Gauss-Newton method"

Hmm, I guess you want to say that sometimes, one may try to find nonlinear feedbac which are optimal. Yes, in very special cases this is possible. However, it remains restricted to very special (or small dimensional) cases.

To sum up, MPC is powerful yet it requires strong assumptions to work properly. However, claiming that "LQR methods are enough" might be a bit too strong (imho).

 Best. 

Hi Timm,

Thank you for your answer. 

About the hessian growing unstable, perhaps I don't know other formulations other than GPC but at least from what I know if you have a state space in the form,

X(k+1)=AX(k)+BU(k)

and your predictions in the form, 

Yfuture=GXk+HUfuture

where, 

G=[CA;CA^2;.......;CA^(Np-1);CA^(Np)];

and 

H=[CB, 0, 0,....,0;

CAB,CB,0,...,0;

CA^2B,CAB,CB,0,....;

....,....,...,...,..;

CA^(Np-1)B,CA^(Np-2)B,.....,CA^2B,CAB,CB];

and your matrix A has one pole outside unit circle, then your hessian...

E=(H'QH+R)

grows unstable for a sufficiently large prediction horizon, which means you can't match the LQR solution EVEN in the linear case. 

Now, ofcourse you can implement some "predescribed degree of stability" using exponentially weighted to avoid this hessian growing unstable (see MPC System Design and Implementation using Matlab, Wang 2009), but then again, we start to go further into modifications of the normal cost function that wont match the LQR solution.

And also, I do think PFC is a predictive controller because its control law is based on future errors. The fact that they impose a basis function for its input, doesnt mean that is not predictive or optimal. It is simply optimal to THAT cost function, not to the original-normal one. Its like saying Laguerre Polynomial are not optimal because you impose the control basis function. Again, to solve it you change your index to find Laguerre coefficients instead of all the input vector, so is optimal to THAT cost function. And PFC is also a receding horizon optimal control. Receding horizon refers simply to the fact that you take the first step of your solution, use it, then "recalculate" your input in the next sampling time using the new state. The recalculation is unnecesary when you are unconstrained and linear time invarying since you can find a control law in the form u(k)=-Kx(k)+Kr Yr. Now for a PFC with the control basis u(k)=u(k+1)=u(k+2)=....., this can be easily done with GPC by just taking the last row of G as a coincident point then summing the rest of the inputs as in, 

G=CA^Np

H=[C(A^(Np-1)+....A^2+A+I)B]

Select a reference trayectory for your coincident point, Yr=alpha y_k+(1-alpha) ref

And you have your control law as, 

u(k)=(H'H)^-1H'(Yr-GXk)

Notice I omitted Q and R weighting which is usually done for PFC to avoid having more decision variables than prediction horizon, reference trajectory and input basis function. 

Now about LQR tracking, if your reference is simply a set point which is usual in many applications, then you can implement integral version state space and thats it (see Ogata Discrete Time Control Systems, 2ed, chapter 8th). But I agree, that if you want to do trayectory tracking, then MPC presents itself as a much powerful solution than LQR, but in any case, if your system is unstable, I must insist in recommending Dual Mode MPC so your solution is not restricted to short prediction horizons where your hessian hasnt grow unstable and you can have some stability guarantees while using MPC simply for performance enhancement or contraints handling.

Regards,

Oscar

Dear Oscar,

I am talking about MPC as receding horizon optimal control, cf. the following books:

• Grüne, L. & Pannek, J. Nonlinear Model Predictive Control: Theory and Algorithms Springer Verlag, 2011
• Rawlings, J. & Mayne, D. Model Predictive Control: Theory & Design

Nob Hill Publishing, Madison, WI, 2009

In MPC (i.e. in receding horizon optimal control as considered in the books cited above) there is no restriction to stable plants.

If you refer to GPC than indeed we are discussing about different control methods, based on diferent assumptions and settings.

Hi Timm, 

I would be very interested in seeing the solution of it for the unconstrained linear case when you have unstable plant for horizons above 100-500.   If you have any code that you could share with me for testing and comparing, I would be extremely grateful.

Here is the Matlab code that I use for getting my MPC unconstraind control laws which I am currently working on implementing in UAVs. I left horizon fixed at 500 just to show you what I mean. 

Now, if you use this system with unstable pole at z=1.040707

A=[1.8147 -0.8055 -0.0076

1 0 0

0 0 0]

B=[0.0559

0

1]

C=[1

0

0]               

Try it with my code Kx=GetMPC(A,B,C,0.1) 

you get, 

Kx=[-897.6918 694.8073 6.555599]

Get LQR, you get Kx=dlqr(A,B,diag([1,0,0]),0.1);

Kx=[8.35234804130849 -5.61622229283201 -0.0529898096667414]

Now, try a stable plant simply by,

A=[1.8147 -0.8247 -0.0076

1 0 0

0 0 0]

You get, 

Kx_MPC=[7.913714 -5.526857 -0.0509326]

Kx_LQR=[7.91372157966594 -5.52686361465108 -0.0509326594961522]

So you can see, when the system is stable it matches with LQR infinite horizon for big horizons, whereas for unstable it doesnt. However, if you select short horizon, say 50 instead of 500, it matches in both cases. Try GetMPC50(A,B,C,0.1).

Sorry, I didnt have time to modify the code for variable prediction horizon. I have it like that for a hardware implementation.

I think this is an important thing to bare in mind when you implement MPC together with adaptive or online system identification schemes because if the identified system presents some of this properties like unstable poles, it may give you an unstable control law. Ofcourse you can restrict the identified plant not to accept unstable poles but if it happens, then you might run into a problem.

Finally I would just like to say, there is no difference in the MPC that I just showed you and the "receding horizon control". Like I said, "receding horizon principle" referes to applying only the first input of your solution at current sampling time, then apply another "first input" in your next sampling time, and so on.

Regards,

Oscar

Hi Kerim, 

I agree that MPC lacks of the robustness of other controllers like H infinity, however, when coped with a PROPER online system identification algorithm that can deal with systems without persistent excitation, noise and unbiased predictions, I think it would easily compete with Hinfinity. And also, in Hinfinity you also use frequency response for achieving your performance, another criteria isnt it? Simply different. Let me ask you this, what do you want to be robust to? noise? input/output disturbances? plant uncertainty? tracking? Each of this has very different requirements in the frequency response. Also, think about it, if you dont know what your plant is how aggressive and robust you think you can be to disturbances?  In my humble opinion I think each has its own merits. Certainly MPC has a lot of parameters to tune that could give a certain frequency responsee, e.g. number of inputs, spacing between inputs (move blocking), predictions horizon, trayectory considerations, number of points to consider (you can consider only certain parts of the trayectory, weighting matrices to penalize parts of the trayectory more and so on. Each of this has a significant impact on the frequency response of the system. If you have certain frequency response in mind, you can use Genetic Algorithms to find the weighting of matrices of the MPC that would give you the desired performance. Also, what about supervisory control? MPC has a model implicit in it, which means you can cope it with multiple model interacting filters to achieve control of systems in different operating points (nonlinear), or even, systems with faults.

As for not being implementable on large systems and fast systems, the Real Time Iteration Scheme makes the online optimization achievable. And if the system is large, a prestored control allocation law can be used to avoid increased number of decision variables.

Regards,

Oscar

Hi,

Here is a complete workshop on how to implement MPC and MHE in MATALB.

https://www.youtube.com/playlist?list=PLK8squHT_Uzej3UCUHjtOtm5X7pMFSgAL

The workshop shows a complete explanation of the implementation with coding examples. The codes are also provided!

Hi, MPC is model prodictive controller, meaning it can predict the next sequence of the control signals based on current states of the system. It is an online controller and comparing to offline optimal strategies such as LQR it can be updated by the real system state other than the model. More accuracy is the result if the plant evolves slow enough to provide the gap for MPC to compute the next cycle. Nonlinear systems which are not easy to be linearized can be an option to this Controller.

Optimal control is an open-loop approach, Model Predictive Control (MPC) is optimal control but it is closed-loop over finite receding horizon time. that make the controller robust and stable

see

Book Model Predictive Control Toolbox

I would like to make an acknowledgement for advanced proces control of Heat exchangers

Judged by all conceivable performance criteria, multivariable neural network predictive control is currently considered to be the best available control technology. And this control technology can only be outperformed by a real feedforward control. Real feedforward control is the technology that completely and directly counteracts the effect of disturbances without first having to wait for the disturbances to affect the process output.

Please continue reading Heat exchanger control: Technical Report Disclosure of elementary feedforward control algorithm for h...

Dear Debanjan Chatterjee,

beyond the previous comments.

MPC considers many properties, such as stability, robustness and disturbances rejection in closed loop. Researchers are always developing new theories and algorithms to improve this control family.

Regards.

Dear Debanjan Chatterjee ,

This is the receding horizon approach. The idea is to update the optimal values in each sampling time interval. The optimization procedure calculates a sequence of optimal control inputs along the prediction horizon considering the current states. However, disturbances and other changes are affecting the processes all the time and, thereby, modifying the optimal operating points. please see the following book for detail:

Zhang, Weidong. Quantitative process control theory. Vol. 45. CRC press, 2011.