CN106933107B - A kind of output tracking Robust Predictive Control method based on the design of multifreedom controlling amount - Google Patents
A kind of output tracking Robust Predictive Control method based on the design of multifreedom controlling amount Download PDFInfo
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Abstract
The invention discloses a kind of output tracking Robust Predictive Control methods based on the design of multifreedom controlling amount, for a kind of smooth nonlinear system, using RBF-ARX modeling method, pick out the system mathematic model comprising system bounded uncertain noises, the design feature for making full use of the model constructs the multiple variable linear polyhedral models for capableing of baggage systems Nonlinear Dynamic.In view of system future control input increment by STATE FEEDBACK CONTROL rate provides the conservative that may cause in the design of single-degree-of-freedom control amount in Infinite horizon objective function, the invention proposes a kind of output tracking Robust Predictive Control methods based on RBF-ARX model based on the design of multifreedom controlling amount.Compared with prior art, the method for the present invention considers the unknown BOUNDED DISTURBANCES of system, is not necessarily to systematic steady state equalization point information in controller design and effectively increases the freedom degree of optimal control increment, has higher robustness and practical value.
Description
Technical Field
The invention belongs to the field of automatic control, and relates to an output tracking robust predictive control method based on multi-degree-of-freedom control quantity design.
Background
Model predictive control is a new class of computer control algorithms that emerged in the field of industrial process control in the 70's of the 20 th century. As a typical representative of the advanced process control, the control mechanism is adaptive to the complex industrial process, so that the advanced process control algorithm is the most favored advanced control algorithm in the industrial process control field and has been applied with a great deal of success. In practical applications, linear model-based predictive control algorithms were first applied to the control of complex systems. However, strictly speaking, almost all practical control systems are nonlinear, and the nonlinear problem existing in the industrial practical application is not well solved by the prediction control based on the linear model. Nonlinear model predictive control is a hotspot of extensive research at present, and a series of theoretical achievements and practical applications are achieved. However, the predictive control algorithm directly using the nonlinear model generally needs to solve a high-order non-convex nonlinear optimization problem with constraints on the belt on line, which usually causes higher calculation cost and even cannot guarantee a certain feasible solution.
In recent years, as the design theory and method of predictive control stability become mature, the robust predictive control research on uncertain objects in complex environments gradually becomes a hot spot of the predictive control research. At this stage, based on the min-max principle, the robust predictive control method is widely researched by using tools such as invariant set theory, Linear Matrix Inequality (LMI) solution and the like. Most proposed robust predictive control methods so far are state regulation or output tracking control methods proposed on the premise that system state operating point information is known. For example, the patent application for the modeling and control method of the nonlinear CSTR system in the prior art includes 'aggregation prediction control system based on model prediction control and control method thereof' (application number: 200910197512.0) 'hybrid model optimization control method of a first-order reaction continuous stirred tank reactor' (application number: 201010616956.6), 'rolling time domain estimation method of a continuous stirred tank reactor with multi-rate sampling' (application number: 201310311184.9) 'integrated multi-model control method of a continuous stirred tank reactor' (application number: 201510315584.6). The technical characteristics of the patent application of the invention are as follows: are system dynamic state space model and controller designs, all under the assumption that the state steady state equilibrium point information of the nonlinear CSTR system is known or given. In this kind of method, it is assumed that the state of the system is fully measurable, however, in practical application, there is a large class of non-linear systems whose steady-state operating point information is unknown or difficult to obtain due to the existence of non-measurable interference or modeling error. Therefore, the output tracking robust predictive control method for the complex nonlinear system with unknown or difficult-to-obtain system balance point information is the main problem to be solved urgently in actual control. The application number '201610139588.8' published in 2016, 8, 24 and provides a robust predictive control algorithm for CSTR system output tracking based on a nonlinear ARX model under the condition that the steady state balance point information of the CSTR system is unknown. However, the controller design of the method does not consider the external interference of the system, and in practical application, the control system is inevitably affected by various complex external uncertain interferences, so that the method is only suitable for the control of the CSTR system with weak external interference or low requirement on control robustness, and has obvious limitation. In addition, when the quadratic objective function of the method is designed, the future control input increment of the system is given by the same state feedback control rate, the method is a single-degree-of-freedom control quantity design method, and the controller design has stronger conservatism.
Disclosure of Invention
The invention aims to provide an output tracking robust predictive control method based on multi-degree-of-freedom control quantity design aiming at the defects in the background technology, the method considers the influence of unknown bounded disturbance of a nonlinear system, effectively increases the degree of freedom of the control quantity in the design of a controller objective function, and has wider applicability, higher robustness and higher practical value.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows:
an output tracking robust prediction control method based on multi-degree-of-freedom control quantity design comprises the following steps:
1) performing off-line identification on the smooth nonlinear system, and establishing an RBF-ARX model containing bounded uncertain disturbance of the nonlinear system:
wherein: y (t) is the output of the nonlinear system at time t; u (t) is the input of the nonlinear system at time t; zeta (t +1) represents bounded uncertain disturbance of the nonlinear system, and | zeta (t +1) | is less than or equal to rho, rho>0 is a known constant; n isyAnd nuRespectively the output and input orders of the model, and kn＝max(ny,nu) (ii) a w (t) is a state variable of the nonlinear system at the moment t; h is the node number of the intermediate layer of the RBF neural network; d is the dimension of the nonlinear system state variable w (t); phi is a0,t，{ak+1,t|k＝0,...,kn-1} and { bk+1,t|k＝0,...,kn1 is a Gaussian-based RBF neural network dependent on state quantities w (t)An autoregressive coefficient;andlinear coefficients of the RBF neural network;as the central vector of the RBF neural network,element values of RBF neural network central vector;a scaling factor of the RBF neural network;is the 2 norm of the vector; the order of the RBF-ARX model in the above formula includes: n isy，nuH and d; the linear and non-linear parameters of the model include:andthe order and the parameters of the RBF-ARX model are obtained by the off-line optimization calculation of the SNPOM optimization method (the SNPOM optimization method is an off-line optimization method combining the Levelenginequintet method (LMM) and the linear Least Square Method (LSM), and the details are Peng H, Ozaki T, Haggan-Ozaki V, Toyoday.2003, A parameter optimization method for the radial basis function types);
2) constructing a plurality of variable linear polyhedral models which wrap the object in a nonlinear dynamic manner and consider the bounded uncertain disturbance of a nonlinear system by using the structural characteristics of the RBF-ARX model;
first, the following system deviation variables are defined:
wherein:the nonlinear system output increment at the time of t + i; y (t + i) is the output quantity of the nonlinear system at the moment of t + i; yr is the desired output of the nonlinear system;the nonlinear system input increment at the moment t + j; u (t + j) is the nonlinear system input quantity at the moment of t + j; u (t + j-1) is the nonlinear system input quantity at the moment of t + j-1; obtaining the output deviation of one-step forward prediction of the t-time model by the above formulaThe following were used:
wherein: ζ (t +1| t) represents a one-step forward prediction of bounded uncertain disturbances of the system, and | ζ (t +1| t) | ≦ ρ;in order to be the intermediate vector, the vector is,similarly, the output deviation of two-step forward prediction of the t-time model is deducedThe following were used:
finally, a plurality of variable linear polyhedral models (state space models) X (t +1| t) for describing the current nonlinear characteristic and the future nonlinear characteristic of the system are obtained, and the structures of X (t +2| t) and X (t + g +1| t) are as follows:
wherein A ist,BtThe RBF-ARX model parameter matrix is obtained by offline identification at the moment t; x (t | t) is an RBF-ARX model state vector obtained by offline identification at the time t;inputting an increment for the nonlinear system at the time t, wherein the increment is a parameter to be optimized; a. thet+1|t,Bt+1|tIs a nonlinear system state matrix at the moment t +1, and X (t +1| t) is a nonlinear system state vector at the moment t + 1;inputting increment for the nonlinear system at the t +1 moment; a. thet+g|t,Bt+g|tThe nonlinear system state matrix at the future time t + g is shown, and X (t + g | t) is a nonlinear system state vector at the future time t + g;the nonlinear system input increment at the moment t + g; xi (t) in the above formula is dynamically wrapped by a convex linear polyhedron as follows:
and is
Wherein xit,s(s is 1,2) is a bounded perturbation term at time t,is a polyhedral time-varying linear coefficient;term represents equal toOrWhen s is equal to 1, is takenWhen s is 2, is takenAnd isIs the upper limit value of the variable ζ (t +1| t), i.e Is the lower limit of the variable ζ (t +1| t), i.e.Similarly, xi (t +1| t) is derived to be dynamically wrapped by a convex linear polyhedron as follows:
and is
Wherein xit+1|t,s(s＝1,2)Is a bounded perturbation term at time t +1,is a polyhedral time-varying linear coefficient;term represents equal toOrWhen s is equal to 1, is takenWhen s is 2, is takenAnd isIs the upper limit value of the variable ζ (t +2| t), i.e. Is the lower limit of the variable ζ (t +2| t), i.e.
State matrix A of system at future time t + gt+g|t,Bt+g|tDynamically wrapped by two convex linear polyhedrons as follows:
wherein:is a polyhedral time-varying linear coefficient, Lh＝2h(ii) a The vertex of the polyhedron is { Ak|k＝1,2,…,LhAnd { B }l|l＝1,2,…,Lh}, wherein: a. thekAnd BlThe elements in the RBF-ARX model are obtained by calculating the upper and lower limit information of the state dependent function coefficient in the RBF-ARX model, and the calculation expression is as follows:
wherein:namely, it isAnd isThe linear coefficients of the RBF neural network in the model can be obtained by off-line identification through an SNPOM method.
In the above formulaItems andthe terms are calculated using the following formulas:
wherein,term represents equal toOre y,mAnd is andas a function of the variable w (t)The upper limit value of (a) is,as a function of the variable w (t)A lower limit value of (d);term represents equal toOre u,mAnd is andas a function of the variable w (t)The upper limit value of (a) is,as a function of the variable w (t)A lower limit value of (d); obtained by the above formulaAndthe interval of change of (a) is,due to AkContains h in the expressionTerm (m-1 … h), each of whichThe values of the terms are 2, so that 2 can be constructedhIs as in formula AkWherein k is 1 … 2h(ii) a In the same way, because BlContains h in the expressionTerm (m-1 … h), each of whichThe values of the terms are 2, so that 2 can be constructedhIs as in formula BlWherein l is 1 … 2h(ii) a Thus, two convex linear polyhedrons Ω are obtainedAAnd ΩB2 of (2)hA vertexAndwherein L ish＝2h；The central vector and the scaling factor of the RBF neural network can be obtained by off-line identification through an SNPOM method;
3) designing a quadratic objective function of an infinite time domain of the system based on the constructed variable linear polyhedron models, and solving a min-max optimization problem to obtain the optimal control quantity of robust predictive control;
based on the variable linear polyhedron models constructed in the above way, the following min-max optimization problem is solved to obtain the optimal prediction control quantity which is robust and stable to the bounded uncertain disturbance:
wherein W is not less than 0, R>0 is a weighting coefficient of the control; in the invention, an infinite time domain quadratic form objective function is adoptedIs divided intoAndthree parts: wherein, in the present invention,respectively contained in Are used as objective functions to optimize the calculated control quantities,included in the system future control input incrementThen given by the state feedback control rate.
Based on a design method of an invariant set, the infinite time domain optimization problem can be converted into a linear programming problem with Linear Matrix Inequality (LMI) constraint for solving convex optimization as follows:
wherein: symbol represents the symmetric structure of the matrix; i represents an identity matrix; w is not less than 0, R>0【W＝1,R＝0.02】；umaxAnd uminRespectively the maximum value and the minimum value of the system input quantity;the maximum value of the system input increment is,the minimum value of the system input increment can be determined by the actual value range of the system input increment; z is a symmetric matrix; f (t) ═ YG-1Feeding back a gain matrix for the system; qklAnd QefIntermediate matrix variables for solving the convex optimization problem, where k, L, e, f is 1,2, …, Lh(ii) a In the above linear matrix inequality, Y, G, Qkl，Z，Andall intermediate variables in the process of solving the minimization variable gamma, and solving the minimization problemThen, the optimization function will automatically find the intermediate variables Y, G, Q that satisfy the minimum γ according to the above constraint conditionskl，Z，Andwhen suitable intermediate variables Y, G, Q are foundkl，Z，Andif so, ending the minimization optimization solving process at the time t;
at each sampling time t, obtaining the optimal system input increment by solving the convex optimization problem of the linear matrix inequalityFurther obtains corresponding optimal control quantity input acting on the nonlinear system as
Compared with the prior art, the invention has the beneficial effects that:
aiming at a class of smooth nonlinear systems, the invention adopts an RBF-ARX modeling method to identify a system mathematical model containing system bounded uncertain interference, and constructs a plurality of variable linear polyhedron models capable of wrapping system nonlinear dynamics by fully utilizing the structural characteristics of the model. In consideration of conservatism caused by the fact that future control input increment of a system in single-degree-of-freedom control quantity design in an infinite time domain objective function is given by a state feedback control rate, the invention provides an output tracking robust prediction control method based on an RBF-ARX model and designed based on multi-degree-of-freedom control quantity. Compared with the prior art, the method considers the influence of external uncertain interference of the system, does not need system steady state balance point information in the design of the controller, effectively increases the degree of freedom of optimizing control increment, and has higher robustness and practical value.
Drawings
FIG. 1 is a schematic diagram of an embodiment of the present invention as applied to a CSTR system.
Detailed Description
A schematic diagram of an embodiment of the present invention applied to a CSTR (continuous stirred reactor) system is shown in FIG. 1. In this embodiment, an exothermic, irreversible reaction takes place in the CSTR system, the reaction raw material a and the product B, the reaction raw material a flows into the reactor at a certain flow rate, and the resulting reaction mass flows out of the reactor at the same flow rate. The relevant parameters in the CSTR system shown in fig. 1 are: the reaction temperature in the reactor isT, coolant temperature TcThe feed concentration of the reaction raw material A is CAf1mol/L, feed rate Qf100L/min, feed temperature Tf350K, the reactor volume V100L, the specific heat Cp0.25J/g.K, the concentration of A after the reaction is CAThe product of the heat transfer coefficient and the reactor surface area is UAh＝5.8×104J/(mingk). In this embodiment, the input to the CSTR system is the coolant temperature TcThe output is the reaction temperature T.
The present invention is illustrated in its embodiments by the CSTR system described above.
1) Acquiring the historical input and output data 2000 points of the CSTR system, and obtaining the RBF-ARX model of the CSTR system by off-line identification, wherein the structure is as follows:
wherein: t (t) is the output of the system at time t; t isc(t) is the input to the system at time t; zeta (t +1) represents the uncertain disturbance of the system with boundedness, and zeta (t +1) | is less than or equal to 5; phi is a0,t，{ak1+1,t|k10, 5, and { b }k2+1,t|k20, 5 is a gaussian RBF network type autoregressive coefficient dependent on the state quantity w (t); selecting a system output as a state quantity, namely w (t) ═ t (t);the central vector of the RBF neural network is obtained;a scaling factor of the RBF neural network;is the 2 norm of the vector;andlinear coefficients of the RBF neural network; nonlinear parameter of RBF-ARX model in formula (1)And linear parameterCan be obtained by off-line optimization calculation through an SNPOM optimization method (see Peng H, Ozakit, Haggan-Ozaki V, Toyoda Y.2003, A parameter optimization method for the radialbasis function types of models); in the embodiment, the nonlinear parameters are obtained by the off-line optimization calculation of the SNPOM optimization methodLinear parameter
2) And constructing a plurality of variable linear polyhedral models which wrap the non-linear dynamic state of the CSTR system and consider the bounded uncertain disturbance of the system by utilizing the structural characteristics of the RBF-ARX model.
The system is defined as the following deviation variables:
wherein:the system output increment at the time t + i; t (T + i) is the system output quantity at the T + i moment; t isrIs the desired output of the system;the system input increment at time t + j; t isc(t + j) is the system input quantity at the time of t + j; t isc(t + j-1) is the system input quantity at the moment of t + j-1; the output deviation of one-step forward prediction of the t-time model can be obtained by the formulaThe following were used:
wherein: ζ (t +1| t) represents a one-step forward prediction of bounded uncertain disturbances of the system, and | ζ (t +1| t) | is less than or equal to 5;in order to be the intermediate vector, the vector is,similarly, the output deviation of two-step forward prediction of the t-time model can be deducedThe following were used:
finally, a plurality of variable linear polyhedral models (state space models) X (t +1| t) for describing the current nonlinear characteristic and the future nonlinear characteristic of the system are obtained, and the structures of X (t +2| t) and X (t + g +1| t) are as follows:
wherein A ist,BtIs a parameter matrix obtained by offline identification at the moment t; x (t | t) is a state vector obtained by offline identification at the moment t;inputting an increment for the system at the time t, wherein the increment is a parameter to be optimized; a. thet+1|t,Bt+1|tThe system state matrix at the moment t +1 is obtained, and X (t +1| t) is a system state vector at the moment t + 1;the system input increment at time t + 1; a. thet+g|t,Bt+g|tFor the system state matrix at a future time t + g, X (t + g | t) is the system state vector at the future time t + g;inputting an increment for the system at the future time t + g; xi (t) in the above formula is dynamically wrapped by a convex linear polyhedron as follows:
and is
Wherein { xit,s1 or 2 is a bounded perturbation term at the moment t;is a polyhedral time-varying linear coefficient, and term represents equal toOrAnd is Similarly, the attainable xi (t +1| t) is dynamically wrapped by a convex linear polyhedron as follows:
and is
Wherein { xit+1|t,s| s ═ 1 or 2} is a bounded perturbation term at time t + 1;is a polyhedral time-varying linear coefficient, and term represents equal toOrAnd is
State matrix A of system at future time t + gt+g|t,Bt+g|tDynamically wrapped by two convex linear polyhedrons as follows:
wherein:is a polyhedral time-varying linear coefficient, and the vertex of the polyhedron is { Ak1,2 and Bl1,2}, wherein: a. thek，BlThe elements in the RBF-ARX model can be obtained by calculation from the upper and lower limit information of the state-dependent function coefficient in the RBF-ARX model, and the calculation expression is as follows:
wherein:namely, it is
Wherein,term represents equal toOr ey,1And is andas a function of the variable w (t)The upper limit value of (a) is,as a function of the variable w (t)A lower limit value of (d);term represents equal toOre u,1And is andas a function of the variable w (t)The upper limit value of (a) is,as a function of the variable w (t)The lower limit value of (2).
3) Based on the plurality of variable linear polyhedron models constructed in the above way, a quadratic objective function of an infinite time domain of the system is designed, and the optimal control quantity of robust predictive control is finally obtained by solving a min-max optimization problem.
Considering the conservatism possibly caused by the fact that the future control input increment of a system in the design of the single degree of freedom control quantity in an infinite time domain objective function is given by the state feedback control rate, the invention designs an output tracking robust prediction control method based on an RBF-ARX model and designed based on the multiple degree of freedom control quantity, and the method obtains the optimal control quantity of a control algorithm by solving the following linear matrix inequality group:
wherein: symbol represents the symmetric structure of the matrix; i represents an identity matrix; w is 1, R is 0.02; t iscmax＝400，Tcmin200 is respectively the maximum value and the minimum value of the system input quantity;inputting the maximum value and the minimum value of the increment for the system; z is a symmetric matrix; f (t) ═ YG-1Feeding back a gain matrix for the system; qklAnd QefIn order to solve the intermediate matrix variable of the convex optimization problem, the values of k, l, e and f are 1 or 2; in the above linear matrix inequality, the coefficient matrix At，Bt，At+1|tAnd Bt+1|tIs the parameter matrix measured at time t; x (t | t) is the state vector that has been measured at time t; y, G, Qkl，Z，Andall intermediate variables in the process of solving the minimization variable gamma, and solving the minimization problemThen, the optimization function will automatically find the intermediate variables Y, G, Q that satisfy the minimum γ according to the above constraint conditionskl，Z，Andwhen suitable intermediate variables Y, G, Q are foundkl，Z，Andand then, ending the minimization optimization solving process at the time t. At each sampling time t, obtaining the optimal system input increment by solving the convex optimization problem of the linear matrix inequalityThe corresponding optimal control quantity acting on the system is input asThereby, by adjusting the coolant temperature T in real timec(t) achieving CSTR system output T (t) tracking the desired output trajectory.
Claims (2)
1. An output tracking robust predictive control method based on multi-degree-of-freedom controlled quantity design is characterized by being applied to a CSTR system, namely a continuous stirred reactor system, and comprising the following steps of:
1) performing off-line identification on the CSTR system, and establishing an RBF-ARX model containing bounded uncertain disturbance of the CSTR system:
wherein: y (t) is the output of the CSTR system at time t; u (t) is the input to the CSTR system at time t; zeta (t +1) represents bounded uncertain disturbance of the CSTR system, and zeta (t +1) | is less than or equal to rho, and rho is a known constant; n isyAnd nuRespectively the output and input orders of the model; w (t) is a state variable of the CSTR system at the time t; h is the node number of the intermediate layer of the RBF neural network; d is the dimension of the CSTR system state variable w (t); phi is a0,t，{ak+1,t|k＝0,...,kn-1} and { bk+1,t|k＝0,...,kn-1 is the gaussian based RBF neural network autoregressive coefficient dependent on the state variable w (t);andlinear coefficients of the RBF neural network;as the central vector of the RBF neural network,element values of RBF neural network central vector;a scaling factor of the RBF neural network;is the 2 norm of the vector; the order of the RBF-ARX model in the above formula includes: n isy，nuH and d; the linear and non-linear parameters of the model include:andthe order and the parameters of the RBF-ARX model are obtained by off-line optimization calculation through an SNPOM optimization method;
2) constructing a plurality of variable linear polyhedral models which wrap the object in a nonlinear dynamic manner and consider bounded uncertain disturbance of a CSTR system by using the structural characteristics of the RBF-ARX model;
first, the following system deviation variables are defined:
wherein:the CSTR system output increment at the time of t + i; y (t + i) is the output quantity of the CSTR system at the moment of t + i; y isrIs the desired output of the CSTR system;the CSTR system input increment at the moment t + j; u (t + j) is the input quantity of the CSTR system at the moment of t + j; u (t + j-1) is the input quantity of the CSTR system at the moment of t + j-1; obtaining the output deviation of one-step forward prediction of the t-time model by the above formulaThe following were used:
wherein: ζ (t +1| t) represents a one-step forward prediction of bounded uncertain disturbances of the CSTR system, and | ζ (t +1| t) | ≦ ρ;in order to be the intermediate vector, the vector is,similarly, the output deviation of two-step forward prediction of the t-time model is deducedThe following were used:
finally, a plurality of variable linear polyhedron models X (t +1| t), X (t +2| t) and X (t + g +1| t) structures for describing the current nonlinear characteristic and the future nonlinear characteristic of the CSTR system are obtained as follows:
wherein A ist,BtIs obtained by offline identification at time tRBF-ARX model parameter matrix; x (t | t) is an RBF-ARX model state vector obtained by offline identification at the time t;inputting an increment for the CSTR system at the time t, wherein the increment is a parameter to be optimized; a. thet+1|t,Bt+1|tThe state matrix of the CSTR system at the moment t +1, and X (t +1| t) is the state vector of the CSTR system at the moment t + 1;the CSTR system input increment at the moment of t + 1; a. thet+g|t,Bt+g|tThe CSTR system state matrix at the future time t + g is shown, and X (t + g | t) is the CSTR system state vector at the future time t + g;the CSTR system input increment at the moment of t + g; xi (t) in the above formula is dynamically wrapped by a convex linear polyhedron as follows:
and is
Wherein xit,s(s is 1,2) is a bounded perturbation term at time t,is a polyhedral time-varying linear coefficient;term represents equal toOrWhen s is equal to 1, is takenWhen s is 2, is takenAnd isIs the upper limit value of the variable ζ (t +1| t),a lower limit value of the variable ζ (t +1| t); similarly, xi (t +1| t) is derived to be dynamically wrapped by a convex linear polyhedron as follows:
and is
Wherein xit+1|t,s(s＝1,2)Is a bounded perturbation term at time t +1,is a polyhedral time-varying linear coefficient;term represents equal toOrWhen s is equal to 1, is takenWhen s is 2, is takenAnd isIs the upper limit value of the variable ζ (t +2| t),a lower limit value of the variable ζ (t +2| t);
state matrix A of CSTR system at future time t + gt+g|t,Bt+g|tDynamically wrapped by two convex linear polyhedrons as follows:
wherein:andis a polyhedral time-varying linear coefficient, Lh＝2h(ii) a The vertex of the polyhedron is { Ak|k＝1,2,...,LhAnd { B }l|l＝1,2,...,Lh}, wherein: a. thek，BlThe elements in the RBF-ARX model are obtained by calculating the upper and lower limit information of the state dependent function coefficient in the RBF-ARX model, and the calculation expression is as follows:
wherein:namely, it is
In the above formulaItems andthe terms are calculated using the following formulas:
wherein,term represents equal toOre y,m，Term represents equal toOre u,m；
3) Designing a quadratic objective function of the infinite time domain of the CSTR system based on the plurality of constructed variable linear polyhedron models, and solving a min-max optimization problem to obtain the optimal control quantity of robust predictive control;
based on the variable linear polyhedron models constructed in the above way, the following min-max optimization problem is solved to obtain the optimal prediction control quantity which is robust and stable to the bounded uncertain disturbance:
wherein W is more than or equal to 0, and R is more than 0 and is a controlled weighting coefficient;
at each sampling time t, obtaining the optimal CSTR system input increment by solving the min-max optimization problemFurther obtains corresponding optimal control quantity input acting on the CSTR system as
2. The output tracking robust predictive control method based on multiple degree of freedom controlled variable design according to claim 1, wherein the step 3) is specifically:
quadratic form objective function of infinite time domainIs divided intoAndthree parts: wherein, will be provided withAndrespectively contained inAndare used as objective functions to optimize the calculated control quantities,included in the CSTR system future control input incrementsGiven by the state feedback control rate;
based on a design method of an invariant set, the optimization problem of the infinite time domain is converted into a linear programming problem with linear matrix inequality constraint, which is solved by convex optimization, as follows:
wherein: symbol represents the symmetric structure of the matrix; i represents an identity matrix; u. ofmaxAnd uminRespectively the maximum value and the minimum value of the input quantity of the CSTR system;the maximum value of the input delta for the CSTR system,the minimum value of the input increment of the CSTR system is determined by the actual value range of the input increment of the CSTR system; z is a symmetric matrix; f (t) ═ YG-1Feeding back a gain matrix for the CSTR system; qklAnd QefIntermediate matrix variables for solving a convex optimization problem, where k, L, e, f is 1,2h(ii) a In the above linear matrix inequality, Y, G, Qkl，Z，Andare all intermediate variables, Z, in the solving process of the minimizing variable gammajjIs the element on the diagonal of the symmetric matrix Z; in solving the minimization problemThen, the optimization function will automatically find the intermediate variables Y, G, Q that satisfy the minimum γ according to the above constraint conditionskl，Z，Andwhen suitable intermediate variables Y, G, Q are foundkl，Z，Andif so, ending the minimization optimization solving process at the time t;
at each sampling time t, obtaining the optimal CSTR system input increment by solving the convex optimization problem of the linear matrix inequalityFurther obtains corresponding optimal control quantity input acting on the CSTR system as
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