ES2259931B1 - PROCEDURE FOR OBTAINING A QUASI-OPTIMAL PREDICTIVE DIGITAL CONTR0L SYSTEM WITH REDUCED COMPUTATIONAL REQUIREMENTS. - Google Patents

Abstract

The invention consists of a method that allows a customized predictive digital control system to be obtained for a specific process control problem in the time domain, with reduced computational requirements. The process begins with an analysis of the control problem, aimed at obtaining the necessary information to define a series of design parameters in a quasi-optimal predictive control algorithm. Once these adjustments have been made, formulas are provided to calculate the computational characteristics (such as number of operations per second and memory required) that the implementation in a digital system requires. With these requirements we proceed to a description of the elements that will implement the algorithm and its interconnections, together with the specifications of each of said elements. A control system adapted to the problem in question is thus obtained.

Description

Procedure to obtain a control system Quasi-optimal predictive digital with computational requirements reduced

Object of the invention

The present invention relates to a procedure that aims to obtain a system Predictive control and required hardware specifications to implement it, for multi-variable problems, which is applicable to both linear and non-linear plants and where the plant models can be as varied as in a typical predictive control, including but not limited to models Pulse response, step response, and other model forms of state space.

The invention is practically applicable in all fields of industry where it is necessary to implement a predictive control with real-time optimization, but small and low cost.

Background of the invention

Predictive control ( Model Predictive Control , hereinafter MPC) is recognized by the process control industry as the most versatile of control alternatives in the time domain (see EF Camacho and C. Bordons, Model Predictive Control. New York : Springer, 1999). MPC allows to control from very simple systems, to those with more complex dynamics, multivariable, nonlinear, and / or with large delays in the plant. It also integrates optimal control with restrictions on inputs and outputs explicitly.

An MPC control is characterized by the structure reflected in Figure 1. The control consists of two fundamental blocks, a dynamic Model (1) of the plant (2) to be controlled, and an Optimizer (3). Using the outputs (4) of the plant (or measurements) in the past that define the state of the plant, and future inputs (5) that are presumed to be applied to the plant, the model can predict future outputs (6) of the plant. These predicted outputs are compared with the reference path (7) (current and future desired values of the plant outputs), obtaining future errors (8). The optimizer's job is to minimize a cost function (9) based on these errors. The cost function often consists of a weighted sum of the squares of the errors. It is also common to include the weighted sum of the squares of the value of the actions (entrances to the plant), to minimize the energy used by the control system. The optimizer selects future inputs to the plant that originate predicted outputs in the Model that minimize cost. Typically there are restrictions (10) on future entries that can be selected, both according to the value of the inputs themselves (11) (maximum and minimum values of each actuator, or physical element that transforms the input order into an interaction on the floor ), as to the value of the outputs (eg, safety limits for the values of the outputs). This procedure is executed every control cycle. At the end of the control cycle, only the inputs calculated for the next instant of time are actually sent to the actuators of the plant; The rest are discarded. In the next control cycle, the process is repeated, updating the status of the Model with the new measurements of the plant outputs obtained. This closes the feedback loop.

While the MPC control has been very successful in the academic world, its implantation in the industrial world is much more limited. The reasons generally cited are greater difficulty obtaining robust mathematical results and stability (although in practice these controls turn out to be very robust and stable), and especially the mathematical complexity that the optimization process requires. The number of operations necessary to calculate the optimal solution is very high, and by therefore it is not easy to obtain real-time results for fast control systems And fundamentally, there is no quantification of computational requirements based on control problem to solve. MPC control is extended widely in the chemical industry, particularly oil, where the slowness of the processes allows long periods of control, and In addition there is no problem in dedicating a computer system Very powerful to control the process. The problem arises when you want to apply this control system to a quick process, and you It requires that the physical implementation be small in size. This situation, extremely frequent in many applications of control (especially in embedded applications) prevents use of a computer system as powerful as it requires the high frequency with which a problem of optimization The absence of a relationship between control problem specifications and the specifications of the digital system needed to solve it in real time makes it difficult the development of a custom system. It is therefore of great interest for the industry the emergence of predictive control algorithms with lower computational requirements, and specifications precise hardware required to be custom implemented efficiently.

They have recently begun to appear strategies that address the problem of implementing MPC control to systems with limited computing power, such as typical microcontrollers So, Bemporad, Morari, Dua and Pistikopoulos describe a strategy consisting of taking advantage that the solution to a quadratic optimization problem with Restrictions is a family of related applications. There is a law related to each possible set of active restrictions (being a  active restriction that effectively limits the value of the solution, so that if said restriction is removed, the cost function could be reduced). This strategy simplifies a lot of arithmetic operations to perform, since applying a related law is much simpler than performing optimization. All the necessary related laws are pre-calculated and Store in a table. However, complexity is seen displaced to data storage and the search for related law necessary at all times, since the tables get to acquire huge proportions. The authors admit that the number of related applications to store grows super-exponentially with the number of present restrictions, what makes this alternative impractical for all real systems except the simplest (very sized reduced).

Description of the invention

The procedure that the invention proposes allows to obtain a quasi-optimal predictive digital control with reduced computational requirements, through an analysis of the control problem focused on particularizing an algorithm of quasi-optimal control, and obtaining the specifications of the hardware needed to run the algorithm efficiently, with simpler hardware than would require control optimum.

More specifically, the control system Quasi-optimal predictive includes optimal control and control auxiliary, the latter computationally simple, combined with way that allow a simplification of the optimization process, maintaining a quality in the characteristics of the control very similar to that provided by pure optimal control.

The analysis of the proposed control problem includes obtaining characteristics and measures that are used by heuristic methods to adjust the control algorithm to trouble.

The design of specifications obtained at the end of the process involves a description of a digital system that, together with the quasi-optimal control algorithm and the values of the previously obtained parameters constitute a complete solution of control, adapted to the particular problem. The specifications of the hardware solve the unknowns of whether it can be implemented, in a real and practical digital system from the point of view industrial, predictive control, and how to carry out such implementation.

The procedure is thus applicable very broad industrial, including, among others:

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Aerospace applications, where a vehicle physical behavior model increases greatly control performance. In applications aerospace is the convenience of small sizes, and Low power consumption, especially on satellites.

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Automotive applications, such as advanced traction and steering controls, which need to be used a dynamic model of the vehicle, and are intended for production en masse (with a very low cost in the system of control).

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Biomedical applications, such as advanced controls for automatic dosing systems insulin, which take into account complex body models human.

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In general, micro- and nano-technologies that require more complex controls than scale systems macroscopic, due to the greater complexity of physics when the scale is extremely small (many simplifications not apply to microscopic scales).

Description of the drawings

To complement the description that is being performing and in order to help a better understanding of the characteristics of the invention, according to a preferred example of practical realization of it, is accompanied as part member of that description, a set of drawings where with illustrative and non-limiting nature, what has been represented next:

Figure 1.- Shows a corresponding scheme to a conventional predictive control, according to the State of the Art reflected in the corresponding section.

Figure 2.- Shows, according to a representation similar to figure 1, the scheme of the procedure that constitutes the object of the present invention.

Figure 3.- Shows the flow chart of the quasi-optimal control of the invention.

Figure 4.- Shows, finally, the architecture of a digital signal processor used in said quasi-optimal control, over which the hardware specifications.

Preferred Embodiment of the Invention

The present procedure starts from the solution to the following control problem: given a system that can be represented by a certain model, get an MPC control with cost function quadratic and linear constraints, which satisfies certain design principles. The implementation in a digital system of MPC control is very poorly studied, being assumed in most cases that "a computer will execute the control algorithm fast enough ", although there are control algorithms that could not be implemented due to lack of said system. The absence of hardware specifications makes it difficult to practical implementation of these controls, as they develop such specifications for each particular case requires a Deep knowledge of Automatic Control, Programming Non-Linear, Algorithmic and Architecture of Computers A methodology is proposed here that leads from MPC control parameters desired (purely mathematical design), to a digital system that allows its physical implementation, focused on obtaining economic systems and limited dimensions. As far as the inventor's knowledge goes, it does not exist in the currently no predictive control system with optimization in real time as a commercial application for embedded systems and / or Small size and low cost, which highlights the need for have the specifications of an economic digital system and effective, able to implement the theoretical controls that are being developed today for many applications Industrial

First, this methodology replaces the Classic MPC control algorithm for a quasi-optimal control (CCO). The CCO will offer a performance statistically very close to that of a pure MPC control, with a much lower computational cost, since the number of operations to be performed is greatly reduced, and Simplify arithmetic.

The fundamental operation of the algorithm of predictive control, minimization (optimization) of a function of typically quadratic cost, it is simple in the absence of restrictions On the contrary, the presence of restrictions It requires using complex mathematical algorithms. Although the variety of optimization methods is very wide, all of them are very sensitive in certain conditions to rounding errors in mathematical operations. Since in each control cycle the optimization problem conditions are different and very variables, the probability that (at some point) the errors of Rounding operations produce an erroneous result is high. Rounding error problems are inherent in the systems digital, and in the case of MPC controls, the strategy usually be to represent the numerical data in the maximum possible precision, typically 64 bits, and use very complex algorithms that reduce the effect of rounding errors. This implies a great number of operations on very large operands (represented with many bits), which are at the same time slower, consume more energy in the processor, and require a more complex processor and more expensive.

The strategy proposed here of a control quasi-optimal faces rounding errors from a point Statistical view: lower accuracy and algorithms of simpler optimization produce correct results in a Most cases To treat cases where the result is not satisfactory, quasi-optimal control employs mechanisms of Detection of such errors. When one of these infrequent errors appear, an auxiliary control mechanism is used a lot less susceptible to numerical errors. This strategy allows implement the control system with a lot of arithmetic minor, resulting in much simpler, cheaper and rapid.

The quasi-optimal control scheme is presented in Figure 2. The auxiliary control block represents a non-predictive control algorithm, such as a proportional-integral control (PI). It can be seen how the outputs (4) of the plant are supplied directly to a second comparator (13), which subtracts them from the reference, using only the values relative to the instant present in the reference path, since the auxiliary control ( 12) does not have prediction of future departures. The error thus obtained (in classical control nomenclature) is supplied to the auxiliary control. This one fulfills two functions. First, it provides an acceptable control action when the optimization algorithm is unable to obtain a solution close to the mathematical optimum, due to its simplifications. Secondly, a check of the proximity between the path of the plant and the reference path can conclude, when both are sufficiently close, that the predictive control is no longer necessary, and that the small error still existing can be effectively eliminated with auxiliary control Failure to perform the optimization process saves a lot of mathematical and traffic operations between the processor and the digital system memory, with the consequent energy savings in that system, especially relevant in portable applications that run on batteries. The auxiliary control to be used is not unique, but depends largely on the plant to be controlled; what is necessary is that it be a robust control, and calculable with few arithmetic operations. This makes it much more robust to rounding errors, while not placing a significant burden on the processor.

An analysis of computer implementation particular control algorithm allows to obtain a limit superior for the operations per second that are necessary. So It is possible to know the memory that the variables used by the algorithm require. These two values are the basis for determine the specifications of a digital system tailored to the application.

A functional architecture is provided Complete to implement control. In combination with the previously obtained specifications, the architecture allows select a series of commercial components to implement the system, and describes its interconnections. Is especially relevant to highlight that the methodology is oriented to obtain a implementable system with technology currently existing in the market, not requiring the development of components specific, thanks to the proposed combination of the algorithm Quasi-optimal with an appropriate selection of its parameters.

This idea has not been implemented so far in practice. Likewise, there is also no procedure that describes how to develop a complete digital control system capable of taking advantage of the characteristics of a quasi-optimal predictive control algorithm. The present invention closes this gap in the state of the art, by providing the description of a complete control system design process that employs this novel
algorithm.

In a specific example of realization of the invention, the existence of an MPC control design is assumed classic for the plant to be controlled, obtained by any of the methods presented in the References. The characteristics of MPC control that are assumed are:

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The plant model is a State space model. The particular cases but very common impulse response model and response model to step can be described as a space model of state.

x [k + 1] = Ax [k] + Bu [k],

y [k + 1] = Cx [k + 1]

where x [k] is the state vector at time k , u is the vector of the entrances to the plant or performances, and is the vector of the exits of the plant, and A, B and C are the state matrices .

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They are considered restrictions linear inputs, states and / or outputs:

A_ {R} T} x [k] \ geq b

where A T R is the transposed constraint matrix.

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The cost function is a quadratic function:

J [k] = \ sum \ limits ^ {H_ {p}} _ {i = 1} ((y [k + i] - y_ {ref} [i]) ^ {T} (y [k + i] - y_ {ref} [i])) + \ sum \ limits ^ {H_ {c}} _ {i = 1} ((u [k + i]) ^ T Ru [k + i])

where H_ {P} and H_ {c} are the prediction horizon and control horizon variables, defined below, and R is a diagonal matrix with the weights for the weighted sum of squares of the actions.

The nomenclature for control data previously determined predictive that you want to implement using The procedure described here is:

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T: cycle duration of control.

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N_ {i}: number of variables of input (actuators).

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N_ {o}: number of variables of output (sensors).

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N_ {s}: number of variables of been used by the plant model.

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H_ {P}: horizon of prediction.

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H_ {c}: horizon of control.

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N_ {R}: number of equations of restriction.

The control algorithm that replaces the classic MPC control is the quasi-optimal control algorithm. The CCO is computerized through a program whose flow chart is presented in Figure 3, and which includes the fundamental steps required by a CCO. The computer program will vary between different control problems, to better adapt to the particularities of the hardware that is finally selected according to the specifications described below (for example, it will explicitly take advantage of multiple arithmetic units in parallel if they exist), but said program perform at least the processes described in Figure
3.

The preferred heuristic process to decide whether a predictive control action is necessary, or on the contrary the auxiliary control is sufficient, it consists in checking if during the last N_ {UE} cycles, each output has experienced variations smaller than a certain threshold. The threshold selection It depends on the auxiliary control (an error or distance to the reference that the auxiliary control is able to correct quickly). The N_ {UE} selection is made according to the number of cycles that lasts a step response without using auxiliary control: 15% of that value. Other decision processes, how could it be check that the cost function experiences a minor variation that a certain threshold during N_ {UE} cycles, can be used if some particular condition of the control problem so recommend, without changing the structure of the invention.

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The determination of whether the result of the optimization is adequate or preferable (necessary in some cases) using auxiliary control is also a process heuristic. It depends largely on the particular implementation of the optimization process. In general, two can be distinguished cases:

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Rounding errors cause a non-recoverable error, such as the interruption of the algorithm when trying to invert a singular matrix.

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The iterative algorithm of optimization has not sufficiently approximated the solution when the control cycle comes to an end and it is necessary to present a value.

The first case is very easily recognizable, since it consists in the appearance of a condition that blocks the optimization and causes it to return an error to the control program. The second case is detectable by the same procedure that is used to define the optimum condition reached in the optimization algorithm itself. Typically this condition is detected by comparison.

u ^ \ leq \ varepsilon_ {1}

where u ^ {1} is the vector of variables to be optimized in i -th iteration, and \ epsilon ^ {1} is a very small value. At the end of the control cycle, the preferred implementation consists of checking whether

u L \ leq \ varepsilon_ {2}, \ hskip0.5cm \ varepsilon_ {2} \ geq \ varepsilon_ {1}

where L represents the last optimization iteration, and ε2 is a small value adjustable for each specific case by essays.

The specification of a hardware capable of implementing real-time control is detailed below. The first requirement is a digital system that has representation of real numbers, that is, some kind of exponential representation (as opposed to representation of integers or fixed point). Typically commercial processors offer floating point (IEEE standard number 754), although in contrast to the classical methodology, accuracies greater than 32 bits (floating point of simple precision) are not necessary, potentially requiring lower precision of having processors that are available. offer them. This is a first important feature, since it reduces the cost of the hardware. The analysis of the optimization algorithms concludes that, in general, all of them employ the scalar product operation very assiduously, since they consist of operations between matrices, with an abundance of multiplications. To achieve high performance in the calculation of scalar products, it is very desirable to specify that the processor has a multiplication-accumulation unit (MAC). For the same reasons two independent memory banks (Harvard architecture) are specified, in order to be able to read in the same cycle the two data that will be multiplied and accumulated below. There is a category of microprocessors that always includes these specifications, because it is precisely aimed at calculating operations such as scalar products. These processors are the Digital Signal Processors (DSP). Therefore, to implement this control application, the necessary computing power and memory requirements will be used by choosing a processor that satisfies them within the category of DSPs. This contrasts with the processors that have classically implemented control systems, also called microcontrollers (due to the very different characteristics of predictive control). The basic architecture of a DSP reflected in Figure 4. This architecture includes the fundamental elements that allow implementing the control in a digital system. The functional blocks are a core (14) for decoding instructions and generating memory addresses, a logical-arithmetic unit (15), two memory banks (16) and (17), each connected to the processor with a bus (18) of addresses and data. In addition, the processor has at least one data bus (19) and at least one instruction bus (20), as well as an input / output bus (21), which is used to communicate with the external: receive measurements from the sensors (22), send values to the actuators (23), and receive commands. Very often, although this depends on the particular application, the inputs and outputs of the plant are analog values, so it will be necessary to use an analog-digital converter on the outputs or measurements before being received by the microprocessor, and a digital converter -Analogical about the inputs or actions provided by the processor, before being received by the actuators.

For the computer program that implements the CCO can make an analysis of the maximum number of operations arithmetic that are necessary in a control cycle. If of the program "Quasi-optimal control algorithm for systems predictive control ", this upper limit is

N max inst = \ left [40N_ {R} N_ {i} + 10N_ {R} N2} {+} + \ frac {5N3 {i} + 30N2 {i} - 35 N_ {i} {3} \ right] _ {mac} + [40N_ {R} + 20N_ {i}] _ {add}

       \ vskip1.000000 \ baselineskip
    

\hskip3.5cm

+ [10N_{R}]_{mul} + [40N_{R} + 5N^{2}_{i} + 15N_{i}]_{div} + [10N_{i}]_{sqrt}

 \ hskip3.5cm 

+ [10N_ {R}] {mul} + [40N_ {R} + 5N2 {+} + 15N_}] {div} + [10N_ {i}] _ {sqrt}

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where each adding in square brackets separately represents MAC operations, add (add), multiplication (mul), division (div), and square root (sqrt).

The program analysis also identifies the memory that is required to store your data and variables. Defining M = N_ {i} H_ {c} and L = N_ {o} H_ {P}, the memory required for variables is:

Mem = N_ {o} + N_ + 5N_ + 6M + 3L + 2M2 + 2LM + N_R + N_R2M2

The specification for the microprocessor requires a computing power P_ {Calc} (measured in FLOPS, operations of floating point per second) higher than the minimum requirement marked by

P_ {Calc} = \ frac {N max} inst {T}

So and according to the above exposed, the procedure allows that, from a description of the technical problem, physical specifications are produced concrete to implement a solution. The specifications produced by this procedure can be satisfied with components that are currently available in the market, assuming therefore a practical and implementable solution in the present. The specifications produced by this procedure they are much less demanding, that is to say it requires less memory and of less operations, than the existing alternatives, giving rise to cheaper solutions when there is an alternative.

Claims (3)

1. Procedure to obtain a quasi-optimal predictive digital control system with reduced computational requirements, in which a dynamic model (1) of the plant (2) to be controlled and an optimizer (3) are used, so that using the Exits (4) of the plant and future entries (5) that are presumed will be applied to it, to predict future exits (6) of the plant, comparing them with the requirement path (7) to obtain errors futures (8), minimizing a cost function (9) based on these errors (8) and taking into account the existence of restrictions (10) on future entries, which can be selected according to the entries themselves (11) ), characterized in that an auxiliary control (12) is provided that represents a non-predictive control algorithm, such as a proportional-integral control, with the particularity that the outputs (4) of the plant (2) are supplied directly to a second comparator (13), that subtracts them from the reference, using only the values relative to the instant present in the reference path, said auxiliary controller (12) not having prediction of future outputs, and so that the error thus obtained is supplied to the auxiliary control (12 ), which provides an acceptable control action when the optimization algorithm is unable to obtain a solution close to the mathematical optimum, due to its simplifications, and also provides a check of the proximity between the plant path and the reference path , which can conclude, when they are close enough, that predictive control is no longer necessary and the small error still existing is effectively eliminated by auxiliary control. 2. Procedure for obtaining a quasi-optimal predictive digital control system with reduced computational requirements, according to claim 1, wherein the specifications of the hardware configured by a digital system tailored to the plant (2) to be controlled include a description of the necessary basic architecture, consisting of a control core (14), at least one logical-arithmetic unit (15) for integers (for example floating point) not necessarily of high precision (for example 32 bits or less), at least two memory banks (16 and 17) with their corresponding address and data buses, and at least one input / output port connected to the interfaces of interaction with the sensors and actuators of the plant, also including a minimum limit of necessary memory and computing power (measured in floating point operations per second) both depending on the computational parameters of the control to be implemented, as well as or a description of the characteristics required in the processor, which are sufficient to characterize the elements that make up the system: a digital signal processor (DSP) and a suitable memory for it, that meet the minimum requirements or limits referred to, said architecture being achievable both through commercially available discrete elements, as well as through integration into a single silicon intellectual property (SIP) chip, this last realization leading to higher performance and a smaller size of the complete control system. 3. Procedure for obtaining a quasi-optimal predictive digital control system with reduced computational requirements, according to previous claims, characterized in that said procedure takes a theoretical optimal predictive control design for a certain plant, which is not implementable in real time or is they do not know the minimum characteristics of a digital system capable of executing it efficiently, and it transforms it into a quasi-optimal predictive control design, for which hardware specifications are described tailored to the problem (attending to minimize the cost, size and consumption of hardware power), specifications that sufficiently describe the elements that make up the digital system to proceed with its physical realization at the industrial level.