Model reference adaptive control method and system for quasi-periodic time-varying nonlinear circuits

Abstract

The present invention relates to a model-referenced adaptive control method for a nonlinear circuit with quasi-periodic time-varying parameters. The method comprises: establishing a quasi-periodic piecewise steady-state system model for the nonlinear circuit; determining the number of subintervals in the controlled system model by identifying the switching moments of the quasi-periodic piecewise steady-state system with unknown parameters; establishing a stable reference model based on the number of subintervals and the proposed stability condition for the quasi-periodic piecewise steady-state system; establishing linear matrix inequality constraints, solving the Lyapunov matrix parameters based on Lyapunov stability theory, and designing different control gain update rates for each state variable of the nonlinear circuit system to update the real-time control variable; ultimately converging the actual control gain to the nominal control gain, so that the state of the controlled system model with unknown parameters tracks the desired state trajectory. The present invention performs control under conditions with interference input, thus achieving higher practicality.

Description

Technical Field

[0001] The invention relates to a model reference adaptive control method for a nonlinear circuit containing quasi-periodic time-varying parameters, and belongs to the field of model adaptive control. Background Art

[0002] The model reference adaptive control method is a type of parallel control concept. As a control method, it has the characteristics of adaptability and robustness. Compared with other control methods such as PID control, the model reference adaptive control method can achieve self-update of control parameters, and can better cope with external disturbances, and can adjust parameters in real time to adapt to changing conditions. Compared with model predictive control, model reference adaptive control requires less prior knowledge of the system and can control the system under the premise of unknown system parameters. The most important feature is that model reference adaptive control can achieve forced convergence of the state of the controlled system to the state trajectory of the desired system. Unlike general stability control, it can make the state quantity of the controlled system transform according to the desired value according to the designer's ideas.

[0003] In order to realize model reference adaptive control, it is necessary to first establish a desired and stable reference system model, and then use the parameter relationship between the controlled system and the reference system to update the control gain. The methods include direct model reference adaptive control and indirect model reference adaptive control: direct model reference adaptive control directly updates the control gain to obtain a real-time updated control quantity, so the control process is relatively "direct"; in contrast, indirect model reference adaptive control does not make the controlled system track the desired model. Since there is an observation model in the middle for connection, the process is relatively "indirect"; but both direct and indirect model reference adaptive control can make the state of the controlled model track the desired state trajectory, and converge quickly and with high convergence accuracy.

[0004] In recent years, model reference adaptive control methods have been widely used in the control of industrial systems. For example, some existing methods, based on a designed reference model, provide the update rate of the longitudinal adaptive control parameters of a vehicle-mounted variable-wingspan unmanned aerial vehicle (UAV). Using a designed controller, the system's performance closely approximates that of the reference model. However, because the selected reference model is still a steady-state system, this method is not applicable when more complex state trajectories are desired. For another example, some existing technologies address common actuator failures in fixed-wing aircraft by designing an integrated fault-tolerant control strategy that combines a multi-model adaptive unknown input observer with a model reference adaptive control method. However, this method considers the controlled system, the fixed-wing aircraft's lateral motion model, to be a time-invariant system, making it inapplicable for more complex controlled models. Summary of the Invention

[0005] The present invention provides a model reference adaptive control method for a nonlinear circuit with quasi-periodic time-varying parameters, aiming to solve at least one of the technical problems existing in the prior art.

[0006] The technical solution of the present invention relates to a model reference adaptive control method, which is applied to a nonlinear circuit with quasi-periodic time-varying parameters. The method according to the present invention comprises the following steps:

[0007] S100, establishing a quasi-periodic piecewise steady-state system model of a nonlinear circuit; the quasi-periodic piecewise steady-state system model does not have a fixed period, and is used as a controlled system model for approximating the nonlinear circuit;

[0008] S200, determining how many subintervals the controlled system model has by identifying the switching moments of the quasi-periodic piecewise steady-state system with unknown parameters;

[0009] S300, according to the number of subintervals and in combination with the proposed stability condition of the quasi-periodic piecewise steady-state system, a corresponding stable reference model based on the quasi-periodic piecewise steady-state system is established;

[0010] S400, then using the reference model parameters to establish linear matrix inequality constraints, based on Lyapunov stability theory, solving to obtain Lyapunov matrix parameters that stabilize the system, and using the Lyapunov matrix parameters to design different control gain update rates for various state quantities of the nonlinear circuit system to update the real-time control quantity;

[0011] S500 : Make the actual control gain finally converge to the nominal control gain, so that the state of the controlled system model with unknown parameters tracks the desired state trajectory.

[0012] Furthermore, in step S100:

[0013] Suppose there is an interval of cycle duration variation [T min ,T max ], where T min Represents the shortest cycle time, T max Represents the longest loop duration;

[0014] Let the duration of each cycle be T i , T i As the length of the subinterval of the quasi-periodic piecewise steady-state system, and T i ∈[T min ,T max ];

[0015] The state equation of the quasi-periodic piecewise steady-state system model with disturbance input and unknown parameters is expressed as follows:

[0016]

[0017] In the formula, the time-varying parameter matrices A(t), B(t), and B w (t) represent the nominal system matrix, input matrix and interference input matrix respectively, and all parameters are unknown; x(t) = [x1(t), x2(t),…, x n (t)], n is the dimension of the system, x(t) is the state quantity of the system; u(t)=[u1(t),u2(t),…,u m (t)], m is the dimension of the control input, u(t) is the control input of the system; w(t)=[w1(t),w2(t),…,w f (t)], f is the dimension of the interference input signal, w(t) is the interference input signal and its parameters are unknown.

[0018] Furthermore, in step S200,

[0019] S210, by detecting whether there is a jump in the identification result within a short time segment to determine the switching time, to obtain the system matrix A i Control matrix B i The corresponding switching time set T s1 and T s2 ;

[0020] S220, the above switching time set T s1 and T s2 Taking the union can obtain all switching time points within a period of time for a quasi-periodic piecewise steady-state system with unknown parameters. In addition, taking the time segments between the time points and collecting the identification quantity information Φ and Γ obtained above, comparing the identification quantities in different intervals to determine whether the system parameters of the two sub-intervals are consistent, and combining the range of each cycle duration to determine the total number of sub-intervals present in each cycle.

[0021] Wherein, the step S210 includes:

[0022] S211. Assume that the control input of the controlled system model satisfies u(t) = u(t-t e ), let t e is a parameter selected in the identification process, let ∈(t)=x(t)–x(t―t e ),t≥t e , let Ξ(t)=[∈(t),∈(t―t s ),…,∈(t―(N―1)t s )], where t s is the sampling interval;

[0023] S212, let Φ be the identification system matrix A i The matrix quantity, and Φ satisfies Ξ(t)=ΦΞ(t―t s ), where t∈[t1+Nt s +t e ,t2], where N represents the sample size, Each time, a small segment of state data is taken online for calculation to collect information about Φ. When the value of Φ undergoes a certain transformation and the magnitude of the transformation exceeds a certain threshold, it is determined that a switch has occurred, and a set of switching moments is generated and recorded as T. s1 ;

[0024] S213, let Γ be the identification control matrix B i The matrix quantity, Each time, a small segment of state data is taken online for calculation to collect information about Γ. When the value of Γ undergoes a certain transformation and the magnitude of the transformation exceeds a certain threshold, it is determined that a switch has occurred, and a set of switching moments is generated and recorded as T. s2 .

[0025] Furthermore, in step S300,

[0026] For each cycle duration T i ∈[T min ,T max ], without considering the control input and disturbance input, the state equation of the quasi-periodic piecewise steady-state system is expressed as follows:

[0027]

[0028] set up is the duration of the kth subinterval in the i-th cycle, which satisfies set up ρ(.) represents the spectral radius of the matrix, λ * is a positive number selected, representing an exponential decay rate; if and only if or And M i When is a diagonalizable matrix, the quasi-periodic piecewise constant system λ * -The index is stable.

[0029] Furthermore, in step S300,

[0030] The system matrix A of all subintervals of the stable reference model is mi The Hurwitz matrix is selected, and the necessary and sufficient conditions for the stability of the quasi-periodic piecewise steady-state system are used to determine whether the designed system meets the exponential stability condition;

[0031] The state equation of the stable reference model is expressed as follows:

[0032]

[0033] Where A m (t), B m (t) are all quasi-periodic time-varying matrices, r(t) is the reference input signal, x m (t) is the state of the reference model; each cycle contains S subintervals. When time t is in the i-th subinterval, i = 1, 2, ..., S, there is A m (t) = A mi , B m (t) = B mi Wherein, when the time is in the i-th subinterval, the switching signal χ i The value of (t) is 1, and the switching signal χ i The value of (t) is 0.

[0034] Furthermore, in step S400,

[0035] According to the system matrix A of the stable reference model mi Substitute into the following linear matrix inequality system to solve the linear matrix inequality system:

[0036]

[0037] P 1,0 ≤μ1P S,S+1 ,

[0038] P i,i―1 ≤μ i P i―1,i ,

[0039]

[0040] Where, P i,i―1 and P i,i+1 are the Lyapunov matrices corresponding to the initial and final moments of the ith subinterval, respectively. The Lyapunov matrix is a symmetric positive definite form that satisfies P i,i-1 ,P i,i+1 >0, i=1,2,…,S.

[0041] Furthermore, in step S400,

[0042] Assume that the corresponding control gain of each state variable x(t), r(t), w(t) is K x (t), K r (t), K w(t), the real-time control quantity is expressed as follows:

[0043] u(t)=K x (t)x(t)+K r (t)r(t)+K w (t)w(t)

[0044] And suppose that there is a set of corresponding nominal control gains in each subinterval (t) and make the parameters satisfy the following matching relationship:

[0045]

[0046] The parameter update rate of the control gain satisfies the following relationship so that the tracking error e(t) eventually converges to 0:

[0047]

[0048] Where, κ xi , κ ri , κ wi Represents the scaling constant corresponding to the state quantities x(t), r(t), and w(t) when the time is in the i-th subinterval, S i is a fixed matrix, is a positive definite matrix, and P(t) is the Lyapunov matrix obtained by solving the linear matrix inequality.

[0049] Furthermore, in step S400,

[0050] The expression of the Lyapunov matrix P(t) corresponding to the k-th subinterval in the i-th cycle is as follows:

[0051]

[0052] The time-varying Lyapunov matrix P(t) is substituted into the parameter update rate, and the obtained control gain update rate is used to solve the nominal control gain corresponding to each subinterval, and then the real-time control amount is obtained, so that the tracking error e(t) converges asymptotically to 0.

[0053] The technical solution of the present invention also relates to a computer-readable storage medium having program instructions stored thereon, and the above-mentioned method is implemented when the program instructions are executed by a processor.

[0054] The technical solution of the present invention also relates to a model reference adaptive control system, which includes a computer device containing the above-mentioned computer-readable storage medium.

[0055] The beneficial effects of the present invention are as follows:

[0056] The model reference adaptive control method of the nonlinear circuit containing quasi-periodic time-varying parameters of the present invention is used to control the state of a controlled system containing unknown parameters to track to the state trajectory of a desired reference system, realizes the asymptotic tracking of the state of the nonlinear circuit containing unknown quasi-periodic time-varying parameters to the state trajectory of the desired quasi-periodic piecewise steady-state system reference model, and realizes the convergence of the control gain to the standard control gain value.

[0057] After constructing a mathematical model of a quasi-periodic piecewise steady-state system, the present invention first identifies the switching moment of the controlled system, determines the total number of subintervals in the controlled system and the switching order of the subintervals through data processing, and then designs a desired reference model using the stability conditions of the quasi-periodic piecewise steady-state system after obtaining this information. Then, the parameter data in the designed reference model is used to solve the Lyapunov matrix that stabilizes the system, designs the Lyapunov function, and designs the update rate of the control gain through the asymptotic convergence condition, thereby obtaining a continuously changing control gain and a real-time control quantity, and finally achieving the tracking error gradually converging to 0. Moreover, when a reference input signal is introduced to meet the continuous excitation condition, the asymptotic convergence of the control gain can also be guaranteed.

[0058] The present invention aims at a nonlinear circuit system containing quasi-periodic time-varying parameters, introduces an interference input signal to represent the uncertainty factors that often exist in actual circuit systems, and performs control under the condition of interference input, making the control method more practical. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 It is a nonlinear circuit structure diagram and parameter corresponding information according to the present invention.

[0060] Figure 2 The present invention is a flowchart for gradually analyzing and controlling a quasi-periodic piecewise steady-state system with unknown parameters.

[0061] Figure 3 It is an information diagram of the identification quantities Φ, Γ corresponding to the system parameters in each subinterval of the quasi-periodic piecewise steady system according to the method of the present invention.

[0062] Figure 4 It is a trajectory diagram of three actual state quantities and a reference state quantity in a corresponding nonlinear circuit system according to the method of the present invention.

[0063] Figure 5 1 is a graph showing the changing trajectories of the tracking errors of the three state quantities in the corresponding nonlinear circuit system according to the method of the present invention.

[0064] Figure 6 It is the convergence status of each parameter in the actual control gain matrix according to the method of the present invention. DETAILED DESCRIPTION

[0065] The following will provide a clear and complete description of the concept, specific structure and technical effects of the present invention in conjunction with the embodiments and drawings to fully understand the purpose, scheme and effects of the present invention.

[0066] It should be noted that, unless otherwise specified, when a feature is referred to as being "fixed" or "connected" to another feature, it may be directly fixed or connected to the other feature, or it may be indirectly fixed or connected to the other feature. The singular forms "a", "said" and "the" used herein are also intended to include the plural forms, unless the context clearly indicates otherwise. In addition, unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art. The terms used in this specification are only for describing specific embodiments and are not intended to limit the invention. The term "and / or" used herein includes any combination of one or more related listed items.

[0067] Should be understood that, although the present disclosure may adopt the term first, second, third etc. to describe various elements, these elements should not be limited to these terms.These terms are only used to distinguish the elements of the same type from each other.For example, without departing from the scope of the present disclosure, the first element may also be referred to as the second element, and similarly, the second element may also be referred to as the first element.The use of any and all examples or exemplary language ("for example", "such as" etc.) provided herein is only intended to better illustrate embodiments of the present invention, and unless otherwise required, will not impose limitations on the scope of the present invention.

[0068] Reference Figures 1 to 6 In some embodiments, a model reference adaptive control method according to the present invention is applied to a nonlinear circuit with quasi-periodic time-varying parameters, and the method includes at least the following steps:

[0069] S100, establishing a quasi-periodic piecewise steady-state system model of a nonlinear circuit; the quasi-periodic piecewise steady-state system model does not have a fixed period, and is used as a controlled system model for approximating the nonlinear circuit;

[0070] S200, determining how many subintervals exist in the controlled system model by identifying the switching moments of the quasi-periodic piecewise steady-state system with unknown parameters;

[0071] S300, according to the number of subintervals and in combination with the proposed stability condition of the quasi-periodic piecewise steady-state system, a corresponding stable reference model based on the quasi-periodic piecewise steady-state system is established;

[0072] S400, then using the reference model parameters to establish linear matrix inequality constraints, based on Lyapunov stability theory, solving to obtain Lyapunov matrix parameters that stabilize the system, and using the Lyapunov matrix parameters to design different control gain update rates for various state quantities of the nonlinear circuit system to update the real-time control quantity;

[0073] S500 : Make the actual control gain finally converge to the nominal control gain, so that the state of the controlled system model with unknown parameters tracks the desired state trajectory.

[0074] The quasi-periodic time-varying system of the present invention is derived from the phenomenon of non-strictly periodic system dynamics, which is widely found in practical engineering applications. This phenomenon is common in aerospace systems, medical systems, power systems, and the like. Because each parameter cycle in actual engineering applications does not strictly follow a fixed periodic cycle, the duration of each cycle may vary to some extent, and this variation is unpredictable and unknown. The quasi-periodic time-varying system of the present invention better meets the requirements of practical engineering applications. Furthermore, because quasi-periodic time-varying characteristics are difficult to accurately model, unknown parameters and interference inputs are inevitably present in practical quasi-periodic time-varying systems such as nonlinear circuits. The present invention uses a quasi-periodic piecewise steady-state system as a reference model. This method is a model-referenced adaptive control method for unknown quasi-periodic time-varying controlled systems and is suitable for solving nonlinear circuit control problems involving quasi-periodic parameters and interference inputs. Using a quasi-periodic piecewise steady-state system as a reference model not only effectively increases design freedom but also takes into account situations involving interference inputs, making the control method more practical in practical applications. Moreover, the switching moment identification method for quasi-periodic piecewise steady-state systems of the present invention can make the identification more accurate by adjusting the parameters of the identification process, further increasing the flexibility of the identification method, and can use data for identification online rather than offline, which is more suitable for actual engineering applications.

[0075] The present invention also analyzes the conditions for the stability of quasi-periodic piecewise steady-state systems and provides multiple sufficient or necessary conditions. On the one hand, it can more intuitively determine whether a quasi-periodic piecewise steady-state system satisfies exponential stability, which helps to ensure its stability as a reference system; on the other hand, it can more quickly solve the Lyapunov matrix that makes the system stable, which can be used to design the parameter update rate, laying the foundation for the design of model reference adaptive control methods based on quasi-periodic piecewise steady-state systems.

[0076] Specific implementation of step S100

[0077] The present invention provides a modeling method for a quasi-periodic piecewise steady-state system with interference input and unknown parameters, thereby obtaining a quasi-periodic piecewise steady-state system model. A quasi-periodic time-varying system refers to a class of systems with properties that are approximately periodic. Unlike general periodic systems, this system does not exhibit strict periodicity. Furthermore, the system's dynamics exhibit a certain cyclical pattern, but does not have a fixed period. Although the system parameters vary over time, each cycle does not have a fixed duration, and the duration of each new cycle cannot be precisely determined. The present invention addresses the modeling problem of quasi-periodic time-varying systems with the above characteristics and proposes a quasi-periodic piecewise steady-state system model.

[0078] Specifically, the method of the present invention adopts the state space method to construct the model of the quasi-periodic piecewise steady-state system. First, it is assumed that there is an interval with a cycle length change, and it is set as [T min ,T max ], where T min Represents the shortest cycle time, T max Represents the longest cycle duration. It should be noted that the above two values do not need to be very accurate. Assume that the duration of each cycle is T i , T i As the length of the subinterval of the quasi-periodic piecewise steady-state system, and satisfying T i ∈[T min ,T max ], the state equation of the quasi-periodic piecewise steady-state system model with disturbance input and unknown parameters can be expressed as follows:

[0079]

[0080] Where, the time-varying parameter matrices A(t), B(t), B w (t) represent the nominal system matrix, input matrix and interference input matrix respectively, and all parameters are unknown; x(t) = [x1(t), x2(t),…, x n (t)], n is the dimension of the system, x(t) is the state quantity of the system, x(t)∈R n ;u(t)=[u1(t),u2(t),…,u m (t)], m is the dimension of the control input, u(t) is the control input of the system, u(t)∈R m ; w(t)=[w1(t),w2(t),…,w f (t)], f is the dimension of the interference input signal, w(t) is the interference input signal and its parameters are unknown, w(t)∈R f .

[0081] The method of the present invention considers an approximate piecewise steady-state system as the actual control object in each subinterval. Specifically, assuming that there are S subintervals in total, the system matrix, control matrix and interference input matrix in the i-th subinterval are A and i ,B i ,B wi , i=1,2,…,S, and let the percentage of each subinterval in a cycle be η i Assuming that the percentage of each subinterval in each cycle remains unchanged during the quasi-periodic time-varying process, and the overall cycle duration varies indefinitely in each new cycle, the duration of each subinterval in the new cycle will also vary indefinitely. Then, the state equation of the quasi-periodic piecewise steady-state system model when it is in the i-th subinterval is as follows:

[0082]

[0083] It should be noted that the above-mentioned modeling method based on quasi-periodic piecewise steady-state systems of the present invention has a certain universality in practice. For quasi-periodic time-varying systems that are commonly found in nonlinear circuits, contain interference inputs and have unknown parameters, this modeling method is more convenient and intuitive.

[0084] In an application embodiment, see Figure 1 and Figure 2 In the model reference adaptive control method of the nonlinear circuit with quasi-periodic time-varying parameters of the present invention, a nonlinear circuit system with quasi-periodic time-varying parameters is modeled, a state space model of the system is determined, and a state space model of the system is determined based on the selected parameters. L ] as a state variable (see Figure 2 The corresponding voltage v1, v2 and current i L ), with U(t) current source as the real-time control quantity, U w (t) The current source is used as the interference input, and the corresponding nonlinear circuit model can be written as the following expression:

[0085]

[0086] Accordingly, the state equation of the nonlinear circuit system can be expressed as:

[0087]

[0088] The quasi-periodic parameter in this system is the nonlinear resistance R v The change of parameters over time can be approximated as a quasi-periodic piecewise constant form. Considering the quasi-periodic time-varying parameters of the system, the duration of each cycle is T. i ∈[T min ,T max], and there is no relationship between the current cycle duration and the previous cycle duration. Each cycle contains a total of S subintervals, i = 1, 2, ..., S, and the percentage of the duration of each subinterval in each cycle remains unchanged.

[0089] Specific implementation of step S200

[0090] See also Figure 3 The method of the present invention determines the number of subintervals of a quasi-periodic piecewise steady-state system with unknown parameters based on a method for identifying switching moments. In practical applications, there are often situations where system parameters are unknown, including the unknown number of subintervals, which makes it impossible to design the desired reference model accordingly. In the existing applications of reference adaptive control of switching system models, in most cases it is assumed that the total number of seed system modes in the system is known, and the exact moment of mode switching is known. However, in practice, the parameters of the system may be unknown, and it is difficult to directly measure the exact switching moment. The method of the present invention adopts a method for identifying the number of subintervals and the switching order to deal with the situation where parameters are unknown.

[0091] The method of the present invention uses data collection to determine the total number of subintervals in an unknown controlled system and the switching order between subintervals. Specifically, the actual controlled system is operated under relatively ideal conditions, and data is collected and processed through calculations to obtain parameter information of the subintervals of the continuously time-varying system.

[0092] First, by identifying the system matrix A of different subintervals i It should be noted that in this process, the system matrix A is not identified. i The specific value of is not used, but the switching moment is determined by detecting whether there is a jump in the identification result within a short period of time. In which, it is assumed that the system operates under the condition of no interference input. When the control input of the controlled system model satisfies u(t)=u(t―t e ), t e is a parameter selected in the identification process, let ∈(t)=x(t)–x(t―t e ),t≥t e , let Ξ(t)=[∈(t),∈(t―t s ),…,∈(t―(N―1)t s )], where t s is the sampling interval, let Φ be the identification system matrix A i The matrix quantity, and Φ satisfies Ξ(t)=ΦΞ(t―t s ), where t∈[t1+Nt s +t e ,t2], where N represents the sample size, and its specific expression is In this way, a small segment of state quantity data is taken online each time for calculation to collect information about Φ. When the value of Φ undergoes a certain change and the magnitude of the change exceeds a certain threshold, it is considered that a switch has occurred. This ensures the influence of errors caused by calculation. The set of switching moments is recorded as T s1 In this process, we can reduce t s ,t e These two parameters are used to reduce the time period of each identification, thereby improving the accuracy of identification.

[0093] Next, the control input signal is changed to a ramp signal. Specifically, let Γ be the identification control matrix B i Similarly, each time the same time segment as the previous step is taken to collect the corresponding data information online, and solve it through calculation It is understandable that in this process the control matrix B is not obtained. i The specific value of is not the identification quantity related to it, and the relationship it satisfies is By observing the formula, we can know that based on the results of the previous step identification, it is judged that the system matrix A i In the constant sub-interval, and when the control input is a ramp signal and t e When it is a fixed quantity, the identification quantity Γ is only controlled by the control matrix B i It will affect the value, so the identification quantity Γ can identify the control matrix B in different subintervals. i , and similarly, a small segment of state quantity data is taken online each time for calculation to collect information on Γ. When the identification quantity Γ undergoes a certain switch and the size of the switch exceeds a certain threshold, it can be determined that a switch has occurred. All the switching moments obtained are collected, and the set of corresponding switching moments is recorded as T s2 .

[0094] The above two switching time sets T s1 and T s2 Taking the union, we can get all the switching time points of the quasi-periodic piecewise steady-state system with unknown parameters within a period of time. These switching time points represent A i or B i At all moments when switching occurs, as well as the time segments between the time points, the identification quantity information Φ,Γ obtained above is collected. The identification quantities in different intervals are compared to determine whether the system parameters of the two sub-intervals are consistent. Combined with the approximate range of the duration of each cycle, the total number of sub-intervals in each cycle is determined.

[0095] In one embodiment, by max, n≥2 collects state quantity information, and based on the above data processing method, continuously calculates the size of the identification quantity Φ,Γ online. Since the time period selected each time is short, the size of the time period is related to t s ,t e The parameters are related. By comparing the size of the identification quantity in adjacent time periods, when the value changes by more than a certain threshold, it is considered that a switch has occurred. Where Φ and Γ correspond to the system matrix A respectively. i and control matrix B i First, the control input is selected as a step signal, which also satisfies the period condition u(t)=u(t―t e ), identify the system matrix A i The time when the switch occurs T s1 , and retain the parameter information of the identification quantity in each sub-interval. Since there may be system matrix A between different sub-intervals i Same as control matrix B i Different situations, so based on this step, further identify the control matrix B i Get the switching time set T s2 , and also retain the identification information of different sub-intervals collected, and finally combine the identification results of these two steps to obtain all the switching moments T in this short period of time s =T s1 ∪T s2 .

[0096] Next, the identification information for all subintervals is listed. By comparing the identification values of different subintervals, the total number of subintervals and the subinterval switching order can be determined. By observing the changes in the subintervals, a cyclic subinterval sequence is found and compared using the approximate time range of the cyclic duration. If there are multiple subinterval sequences that meet the conditions, the percentage of each subinterval in each subinterval sequence is observed to see if it remains unchanged. Since the percentage of subintervals is unknown, it is necessary to observe whether the percentage of the duration of each subinterval in each new cycle has changed. If there are still several subinterval sequences that meet the above conditions, the smallest cyclic sequence is selected as the actual subinterval switching sequence.

[0097] The method of the present invention identifies the switching moment of the completed mathematical model. This process helps to update and control the parameters of the controlled system under the condition of unknown parameters in the subsequent model reference adaptive control. Therefore, the identification result in this step does not need to be the precise value of the parameter, but rather determines the total number of sub-intervals in the unknown controlled system through information such as the switching moment and the identification amount. This step is performed under the premise that the designer has little prior knowledge of the system.

[0098] Specific implementation of step S300

[0099] The method of the present invention constructs exponential stability conditions for quasi-periodic piecewise-timed systems based on the Lyapunov method, thereby establishing a corresponding desired stable reference model based on the quasi-periodic piecewise-timed system. Based on the properties of the quasi-periodic piecewise-timed system, stability conditions are proposed to ensure that it satisfies exponential stability, including sufficient and necessary conditions. A stable reference model for the quasi-periodic piecewise-timed system corresponding to the number of subintervals of the controlled system is established.

[0100] Among them, for each cycle length T i ∈[T min ,T max ], without considering the control input and disturbance input, the state equation of the quasi-periodic piecewise steady-state system is expressed as follows:

[0101]

[0102] set up is the duration of the kth subinterval in the i-th cycle, which satisfies set up If and only if or And M i When the matrix is diagonalizable, the quasi-periodic piecewise constant system λ can be derived. * -exponentially stable; where λ * is a positive number selected, representing the exponential decay rate; ρ(.) represents the spectral radius of the matrix. Conversely, when the system satisfies λ * -When the index is stable, we can get It can be understood that the above sufficient and necessary conditions can more intuitively determine whether a quasi-periodic piecewise steady-state system satisfies exponential stability.

[0103] Among them, the system matrix A of all subintervals of the stable reference model is mi The Hurwitz matrix is selected. The sufficient condition for the stability of the quasi-periodic piecewise steady system can be used to determine whether the designed system meets the exponential stability condition. The state equation of the stable reference model is expressed as follows:

[0104]

[0105] Where A m (t), B m (t) are all periodic time-varying matrices, r(t) is the reference input signal, x m(t) is the state of the reference model. In the stable reference model, the length of each cycle may vary but contains S subintervals. When time t is in the i-th subinterval, i = 1, 2, ..., S, there is A m (t) = A mi , B m (t) = B mi ; Assume that there is a switching signal χ corresponding to the i-th subinterval i (t), when the time is in the subinterval, its value is 1, and otherwise it is 0. Assume that in each subinterval there is a set of nominal control gains (t) The following parameter matching conditions are met:

[0106]

[0107] It can be understood that the method of the present invention designs a reference model accordingly after determining the number of sub-intervals and the switching order of the unknown controlled system. According to the above-mentioned stability conditions of the quasi-periodic piecewise steady-state system, the same or different reference models are designed for each sub-interval to achieve the desired state trajectory. Therefore, when it is confirmed that the designed reference model meets the stability conditions and has the desired state trajectory, it can be inferred that the designed reference model is reasonable.

[0108] Specific implementation of step S400

[0109] The method of the present invention achieves asymptotic convergence of the state tracking error of the controlled system through a direct model reference adaptive control method. The direct model reference adaptive control method utilizes data from various state variables to obtain a real-time updated control gain update rate, thereby solving for the real-time updated control gain and applying a state feedback control method to obtain an updated real-time control variable.

[0110] First, a stable reference model is established. This is the stable reference model of the quasi-periodic piecewise steady-state system. The update rate of the control gain can be solved by the relationship between the state variables of the controlled system model and the stable reference model. According to step S300 above, the state equation of the stable reference model is expressed as follows:

[0111]

[0112] Where x m (t) is the expected state trajectory of the reference model, r(t) is the input signal of the reference model; when the time is in the i-th subinterval, A m (t) = A mi , B m (t) = B mi , assuming there is a switching signal χ i(t), when the time is in the i-th subinterval, its value is 1, and otherwise it is 0.

[0113] Among them, the tracking error e(t) can converge to 0 asymptotically when time approaches ∞. Since state feedback is used, each state quantity x(t), r(t), and w(t) has a corresponding control gain K. x (t), K r (t), K w (t), the real-time control quantity is expressed as follows:

[0114] u(t)=K x (t)x(t)+K r (t)r(t)+K w (t)w(t)

[0115] And assume that there is a set of corresponding nominal control gains in each subinterval Make sure that the parameters meet the following matching relationship:

[0116]

[0117] When the update rate of the control gain satisfies the following relationship, the tracking error e(t) can be finally converged to 0:

[0118]

[0119] Where, κ xi , κ ri , κ wi Represents the scaling constant corresponding to the state quantities x(t), r(t), and w(t) when the time is in the i-th subinterval. It can be set by yourself, i = 1, 2, ..., S; S i is a fixed matrix that needs to satisfy is a positive definite matrix, and P(t) is the Lyapunov matrix obtained by solving the linear matrix inequality. Under such an update rate condition, it is possible to achieve the result that e(t) can eventually converge to 0 when time t approaches ∞, and the convergence speed is relatively fast.

[0120] Among them, in order to solve the Lyapunov matrix that makes the system stable, another sufficient condition is needed. Specifically, the model reference adaptive control method is designed according to the established reference system model, and the system matrix A of the designed stable reference model is mi Substitute into the following linear matrix inequality system to solve the linear matrix inequality system:

[0121]

[0122] P 1,0≤μ1P S,S+1 ,

[0123] P i,i―1 ≤μ i P i―1,i ,

[0124]

[0125] Where, P i,i―1 and P i,i+1 are the Lyapunov matrices corresponding to the initial and final moments of the ith subinterval, respectively. Furthermore, the Lyapunov matrix is a symmetric positive definite form that satisfies P i,i-1 ,P i,i+1 >0, i=1,2,…,S. It can be understood that when λ i ◇ When it is a positive number, it corresponds to A i is the Hurwitz matrix, otherwise, it corresponds to A i is a non-Hurwitz matrix. Assume that when all subintervals are switched during the solution process, μ i ≥1,i=1,2,…,S, when A mi Choose the Hurwitz matrix, corresponding to λ i ◇ is a positive number, and the expression of the Lyapunov matrix P(t) corresponding to the k-th subinterval in the i-th cycle is as follows:

[0126]

[0127] Solve to get the Lyapunov matrix P corresponding to each subinterval switching moment i,i―1 , P i,i+1 However, since the duration of each cycle will change, the duration of the subinterval will also change accordingly, so we can get the time-varying Lyapunov matrix P(t), and then substitute it into the following parameter update rate:

[0128]

[0129] The control gain update rate can be used to solve the nominal control gain corresponding to each subinterval, and then the real-time control quantity can be obtained to achieve the asymptotic convergence of the tracking error e(t) to 0. In order to facilitate the solution of actual nonlinear circuits and obtain better asymptotic convergence effect, the parameter μ can be taken as i =1,i=1,2,…,S.

[0130] The method of the present invention combines the Lyapunov matrix expression and solves the above-established linear matrix inequality to obtain the Lyapunov matrix that makes the system stable. If the solution can obtain the result, the quasi-periodic piecewise constant system can be obtained to satisfy λ* -Exponential stability. Compared with the above stability conditions, the method of the present invention is not so intuitive in determining the stability of the system, but it can more clearly obtain the Lyapunov matrix that makes the system stable.

[0131] Specific implementation of step S500

[0132] The method of the present invention achieves the asymptotic convergence of the actual control gain to the nominal control gain by introducing a continuous excitation condition for the reference input signal. When the input signal of the stable parameter model satisfies the continuous excitation condition and other conditions remain unchanged, or when the frequency components are sufficiently rich, the tracking error e(t) can converge to 0 as time approaches infinity, while the actual control gain K can be achieved under the condition that other conditions remain unchanged, including the parameter update rate and the designed reference model. x (t), K r (t), K w (t) Converges to the nominal control gain

[0133] Specifically, when the reference input signal introduces a continuous excitation condition, the control gain K(t) can be further realized to converge asymptotically to the nominal control gain K based on the previous step. * (t), this step can increase the robustness of the control process. When the number of bilateral spectrum components of the reference input signal is greater than or equal to n+1, where n is the dimension of the state quantity, when this condition is met, the matrix composed of the state quantity and the control input interference input will satisfy the row full rank, and the control gain K(t) will eventually converge to the unique solution K * (t).

[0134] The present invention realizes the state tracking of a nonlinear circuit system containing unknown quasi-periodic time-varying parameters to the state trajectory of a desired quasi-periodic piecewise constant reference model system. At the same time, when the condition of continuous excitation of the reference input signal is introduced, the control gain can also be converged to the nominal control gain, thereby increasing the system's anti-interference ability and robustness.

[0135] Model Reference Adaptive Control Method and System Simulation Example

[0136] See also Figure 4 、 Figure 5 and Figure 6 The model reference adaptive control method and system of the present invention for a nonlinear circuit with quasi-periodic time-varying parameters are actually tested, wherein a nonlinear circuit system with quasi-periodic time-varying characteristics is considered, the time unit is second (s), and the parameter of each cycle length is set to T i∈[10s,15s], C1=1 / 9F,C2=1 / 24F,R=3Ω,R0=1Ω,L=1 / 7H,there are 3 subintervals in this circuit system, and the percentage of time occupied by each subinterval is η1=0.3,η2=0.2,η3=0.5 respectively. The values of nonlinear conductance in the three subintervals are G1=-0.8S,G2=0.5S,G3=1S respectively. Figure 5 It can be seen that the tracking errors corresponding to each state quantity first converge quickly to a smaller range, and when the time is longer, the tracking error is close to 0. And, see Figure 6 The actual control gain matrix of each parameter convergence situation, taking the control gain matrix of each parameter in sub-interval 3 as an actual example, from Figure 6 It can be seen that when the time is large, the control gain has almost completed convergence.

[0137] It should be appreciated that the method steps in the embodiments of the present invention can be implemented or executed by computer hardware, a combination of hardware and software, or by computer instructions stored in a non-transitory computer-readable memory. The method can use standard programming techniques. Each program can be implemented in a high-level procedural or object-oriented programming language to communicate with the computer system. However, if desired, the program can be implemented in assembly or machine language. In any case, the language can be a compiled or interpreted language. In addition, for this purpose, the program can be run on a programmed application-specific integrated circuit.

[0138] Furthermore, the operations of the processes described herein may be performed in any suitable order unless otherwise indicated herein or otherwise clearly contradicted by the context. The processes described herein (or variations and / or combinations thereof) may be performed under the control of one or more computer systems configured with executable instructions and may be implemented as code (e.g., executable instructions, one or more computer programs, or one or more applications) that is executed collectively on one or more processors, by hardware, or a combination thereof. The computer program includes a plurality of instructions that can be executed by one or more processors.

[0139] Further, the method can be implemented in any type of computing platform that is operably connected to a suitable computer, including but not limited to a personal computer, a minicomputer, a mainframe, a workstation, a network or distributed computing environment, a separate or integrated computer platform, or in communication with a charged particle tool or other imaging device, etc. Various aspects of the present invention can be implemented as machine-readable code stored on a non-transitory storage medium or device, whether removable or integrated into a computing platform, such as a hard disk, an optical read and / or write storage medium, an RSM, a ROM, etc., so that it can be read by a programmable computer, and when the storage medium or device is read by the computer, it can be used to configure and operate the computer to perform the process described herein. In addition, the machine-readable code, or portions thereof, can be transmitted over a wired or wireless network. When such media includes instructions or programs that implement the steps described above in conjunction with a microprocessor or other data processor, the invention described herein includes these and other different types of non-transitory computer-readable storage media. When programmed according to the methods and techniques of the present invention, the present invention can also include the computer itself.

[0140] The computer program can be applied to input data to perform the functions described herein, thereby converting the input data to generate output data that is stored in a non-volatile memory. The output information can also be applied to one or more output devices such as a display. In a preferred embodiment of the present invention, the converted data represents a physical and tangible object, including a specific visual depiction of the physical and tangible object produced on the display.

[0141] The above description is merely a preferred embodiment of the present invention. The present invention is not limited to the aforementioned embodiments. As long as the technical effects of the present invention are achieved by the same means, any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention. Within the scope of protection of the present invention, various modifications and variations of the technical solutions and / or implementation methods are possible.

Claims

1. A model reference adaptive control method, applied to a nonlinear circuit with quasi-periodic time-varying parameters, characterized in that: The method comprises the following steps: S100, establishing a quasi-periodic piecewise steady-state system model of a nonlinear circuit; the quasi-periodic piecewise steady-state system model does not have a fixed period, and is used as a controlled system model for approximating the nonlinear circuit; S200, determining how many subintervals the controlled system model has by identifying the switching moments of the quasi-periodic piecewise steady-state system with unknown parameters; S300, according to the number of subintervals and in combination with the proposed stability condition of the quasi-periodic piecewise steady-state system, a corresponding stable reference model based on the quasi-periodic piecewise steady-state system is established; S400, then using the reference model parameters to establish linear matrix inequality constraints, based on Lyapunov stability theory, solving to obtain Lyapunov matrix parameters that stabilize the system, and using the Lyapunov matrix parameters to design different control gain update rates for various state quantities of the nonlinear circuit system to update the real-time control quantity; S500, making the actual control gain finally converge to the nominal control gain, so that the state of the controlled system model with unknown parameters tracks the desired state trajectory; Wherein, the step S200 includes: S210, by detecting whether there is a jump in the identification result within a short time segment to determine the switching time, to obtain the system matrix A i Control matrix B i The corresponding switching time set T s1 and T s2 ; S220, the above switching time set T s1 and T s2 Taking the union can obtain all the switching time points of the quasi-periodic piecewise steady-state system with unknown parameters within a period of time. In addition, taking the time segments between the time points and collecting the identification quantity information Φ and Γ obtained above, comparing the identification quantities in different intervals to determine whether the system parameters of the two sub-intervals are consistent, and combining the range of each cycle duration to determine the total number of sub-intervals in each cycle; Wherein, the step S210 includes: S211. Assume that the control input of the controlled system model satisfies u(t) = u(tt e ), let t e is a parameter selected in the identification process, let ∈(t)=x(t)–x(tt e ),t≥t e , let Ξ(t)=[∈(t),∈(tt s ),…,∈(t-(N-1)t s )], where t s is the sampling interval; S212, let Φ be the identification system matrix A i The matrix quantity, and Φ satisfies Ξ(t)=ΦΞ(tt s ), where t∈[t1+Nt s +t e ,t2], where N represents the sample size, Each time, a small segment of state data is taken online for calculation to collect information about Φ. When the value of Φ undergoes a certain transformation and the magnitude of the transformation exceeds a certain threshold, it is determined that a switch has occurred, and a set of switching moments is generated and recorded as T. s1 ; S213, let Γ be the identification control matrix B i The matrix quantity, Each time, a small segment of state data is taken online for calculation to collect information about Γ. When the value of Γ undergoes a certain transformation and the magnitude of the transformation exceeds a certain threshold, it is determined that a switch has occurred, and a set of switching moments is generated and recorded as T. s2 .

2. The method according to claim 1, characterized in that In step S100: Suppose there is an interval of cycle duration [T min ,T max ], where T min Represents the shortest cycle time, T max Represents the longest loop duration; Let the duration of each cycle be T i , T i As the length of the subinterval of the quasi-periodic piecewise steady-state system, and T i ∈[T min ,T max ]; The state equation of the quasi-periodic piecewise steady-state system model with disturbance input and unknown parameters is expressed as follows: In the formula, the time-varying parameter matrices A(t), B(t), and T w (t) represent the nominal system matrix, input matrix and interference input matrix respectively, and all parameters are unknown; x(t) = [x1(t), x2(t),…, x n (t)], n is the dimension of the system, x(t) is the state quantity of the system; u(t)=[u1(t),u2(t),…,u m (t)], m is the dimension of the control input, u(t) is the control input of the system; w(t)=[w1(t),w2(t),…,w f (t)], f is the dimension of the interference input signal, w(t) is the interference input signal and its parameters are unknown.

3. The method according to claim 2, characterized in that In the step S300, For each cycle duration T i ∈[T min ,T max ], without considering the control input and disturbance input, the state equation of the quasi-periodic piecewise steady-state system is expressed as follows: x(t)=A i x(t),i=1,2,…,S set up is the duration of the kth subinterval in the i-th cycle, which satisfies set up ρ(.) represents the spectral radius of the matrix, λ * is a positive number selected, representing an exponential decay rate; if and only if or And M i When is a diagonalizable matrix, the quasi-periodic piecewise constant system λ * -The index is stable.

4. The method according to claim 3, characterized in that In the step S300, The system matrix A of all subintervals of the stable reference model is mi The Hurwitz matrix is selected, and the necessary and sufficient conditions for the stability of the quasi-periodic piecewise-steady system are proposed to determine whether the designed system meets the exponential stability condition, so as to ensure the stability of the quasi-periodic piecewise-steady reference system model. The state equation of the stable reference model is expressed as follows: Where A m (t), B m (t) are all quasi-periodic time-varying matrices, r(t) is the reference input signal, x m (t) is the state of the reference model; each cycle contains S subintervals. When time t is in the i-th subinterval, i = 1, 2, ..., S, there is A m (t) = A mi , B m (t) = B mi Wherein, when the time is in the i-th subinterval, the switching signal χ i The value of (t) is 1, and the switching signal χ i The value of (t) is 0.

5. The method according to claim 4, characterized in that In the step S400, According to the system matrix A of the stable reference model mi Substitute into the following linear matrix inequality system to solve the linear matrix inequality system: Where, P i,i-1 and P i,i+1 are the Lyapunov matrices corresponding to the initial and final moments of the ith subinterval, respectively. The Lyapunov matrix is a symmetric positive definite form that satisfies P i,i-1 ,P i,i+1 >0, i=1,2,…,S.

6. The method according to claim 5, characterized in that In the step S400, Assume that the corresponding control gain of each state variable x(t), r(t), w(t) is K x (t), K r (t), K w (t), the real-time control quantity is expressed as follows: u(t)=K x (t)x(t)+K r (t)r(t)+K w (t)w(t) And suppose that there is a set of corresponding nominal control gains in each subinterval And the parameters must satisfy the following matching relationship: The parameter update rate of the control gain satisfies the following relationship so that the tracking error e(t) eventually converges to 0: Where, κ xi , κ ri , κ wi Represents the scaling constant corresponding to the state quantities x(t), r(t), and w(t) when the time is in the i-th subinterval, S i is a fixed matrix, is a positive definite matrix, i = 1, 2, …, S, and P(t) is the Lyapunov matrix obtained by solving the linear matrix inequality.

7. The method according to claim 6, characterized in that In the step S400, The expression of the Lyapunov matrix P(t) corresponding to the k-th subinterval in the i-th cycle is as follows: The time-varying Lyapunov matrix P(t) is substituted into the parameter update rate, and the obtained control gain update rate is used to solve the nominal control gain corresponding to each subinterval, and then the real-time control amount is obtained, so that the tracking error e(t) converges asymptotically to 0. 8 . A computer-readable storage medium having program instructions stored thereon, wherein the program instructions are executed by a processor to implement the method according to claim 1 .

9. A model reference adaptive control system, characterized in that include: A computer device comprising the computer-readable storage medium according to claim 8.

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