CN114265311A - Control method of nonlinear liquid level control resonant circuit system based on dynamic feedback - Google Patents
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Abstract
The invention particularly relates to a control method of a nonlinear liquid level control resonant circuit system based on dynamic feedback. Firstly, determining a mathematical model of a nonlinear liquid level control resonant circuit system; secondly, converting the mathematical model into a state space expression form with input matching uncertainty by selecting a proper state; then, a new state is introduced by utilizing integral control, and the input matching uncertainty is transferred to an initial value of a newly added state through state transformation, so that a state feedback controller is provided; an extended low gain observer with additional matching uncertainty estimate is then designed and an output feedback controller is presented. And finally, verifying the asymptotic stability of the closed-loop system by using a Lyapunov function method. The controller can effectively control the nonlinear liquid level control resonant circuit system, so that all signals of the closed loop system are gradually converged.
Description
Technical Field
The invention belongs to the field of control theory and control engineering, is applied to integral control and self-adaptive control of a nonlinear system, and particularly relates to a control method of a nonlinear liquid level control resonant circuit system based on dynamic feedback.
Background
The linear control theory is the most mature and basic component branch in the system and the control theory, is the cornerstone of the modern control theory, and the linear system has been studied deeply by many scholars in the past decades to form a more perfect control system, but the research is based on: the object is linear, constant and well known, and only so can it be analyzed and controller designed, so it must be known, whether using a frequency domain approach or a state space approach. However, in long-term engineering practice, it is gradually recognized that most of the mathematical models of actual controlled systems are difficult to be known in advance through mechanism modeling or off-line system identification, or some parameters or structures of the mathematical models of the controlled systems are in change, and the systems are also subjected to various disturbances, i.e., the controlled systems cannot completely know — information of uncertainty exists in the systems. Almost all practical systems (chemical, aerospace, artificial intelligence, etc.) suffer from different uncertainties and nonlinearities. While these non-linear parts describe the system structure and dynamic relationship precisely, they also bring great difficulty to the design and stability analysis of the system controller, and at the same time, due to the complexity and diversity of the non-linear system itself, the control scheme thereof has not formed a complete theoretical system so far. Therefore, it is a very significant research topic to study the control of the nonlinear system, both theoretically and practically.
The adaptive technology is one of effective methods for researching nonlinear systems, and the main problem to be solved is to design a satisfactory control system so as to actively adapt to the situations where the characteristics of the system are unknown or changed. In daily life, self-adaptation refers to a feature in which living beings change their habits to adapt to new environments. The basic idea of adaptive control in system design is as follows: in the operation process of the control system, the system continuously measures information such as input, output, state, performance, parameters and the like of a controlled system, so that the current operation index of the system is known and mastered and is compared with an expected index, and then a decision is made to change the structure and the parameters of a controller or change the control action according to a self-adaptive rule so as to ensure the optimal or suboptimal state of the system operation in a certain sense, and the idea of 'dynamic braking' is reflected. It can be seen that the adaptive control relies on less a priori knowledge about the model and the disturbance, which automatically refines the model.
On the other hand, state feedback and output feedback are two main feedback strategies in control system design, and the significance of the feedback strategies is that feedback laws are formed by taking observed states and outputs as feedback quantities, closed-loop control over the system is achieved, and expected requirements for system performance indexes are met. In the state space analysis method of the modern control theory, because the adopted model is a state space model, compared with an output variable, the state variable can completely reflect the internal dynamic characteristics of the system, and the provided information is richer and more comprehensive, so that the state feedback is mostly considered. However, in actual production life, all states of the system cannot be measured for economic reasons and the like, and it is necessary to construct an observer to estimate the state of the system and design a controller based on the observer and the system output. In a word, the method has important theoretical and practical significance as two basic strategies, namely state feedback and output feedback, in control design, and is worthy of further research.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a control method of a nonlinear liquid level control resonance circuit system based on dynamic feedback.
A control method of a nonlinear liquid level control resonance circuit system based on dynamic feedback specifically comprises the following steps:
step 1: establishing a dynamic model of the nonlinear liquid level control resonant circuit system, selecting a proper state to establish state space expression:
in the modeling process of step 1, the influence of deformation of some circuit elements in the nonlinear liquid level control resonant circuit system on the system model is not considered, and the following dynamic description equation of the nonlinear liquid level control resonant circuit system can be obtained:
wherein u represents the control input voltage, VCRepresenting the voltage across the capacitor C, the resistances being R1,R2Inductors are respectively L1,L2The current through the inductor is iL1,iL2The current through the tunnel diode is id＝0.5sin(VC)。
Is to iL1The derivation is carried out by the derivation,
is to VCThe derivation is carried out by the derivation,
is to iL2And (6) derivation.
The system parameters are as follows:
table 1 defines x simultaneously1＝iL1,x2＝-VC,x3＝-0.5(iL2-0.5sin(VC) System (1) is transformed into the following state space representation form:
wherein w is 0.5iL2+0.125(iL2-0.5sin(VC))cos(VC) 0.5u-1 represents the control input, w *1 represents the input match uncertainty, which is considered due to the fact that: in some cases, the control value associated with the desired balance point may not be known, such as in the case of set point adjustment or in the presence of sensor disturbances. And then, based on a dynamic feedback control method, respectively designing a state feedback controller and an output feedback controller for the system (2) to realize the gradual stabilization of the closed-loop system.
Step 2: designing a state feedback controller:
the state feedback controller is designed in the following form:
in the formula, ki> 0, i ═ 0,1,2,3, and are Hurwitz polynomials s4+k0s3+k1s2+k2s+k3R is a constant gain;
and step 3: design of output feedback controller:
the system (2) sets the output y as x1＝iL2And only the output is measurable, the input and output information of the original system is utilized to reconstruct the system state, and the input matching uncertainty is subjected to self-adaptive estimation, so that the influence of the input matching uncertainty is eliminated. The observer was designed as follows:
the control input w is defined as follows:
wherein a isi> 0, i ═ 1,2,3,4, and are Hurwitz polynomials s4+a1s3+a2s2+a3s+a4Coefficient of (d), same hi> 0, i ═ 1,2,3, and is a Hurwitz polynomial s3+h1s2+h2s+h3L is a constant gain.
Preferably, the method for determining the constant gain r specifically includes:
introduction 1: matrix B0
Because B0Is a Hurwitz matrix, so that there is a suitable ki> 0, i ═ 0,1,2,3 and positive definite matrix Q ═ QTSatisfies the following conditions:
B0 TQ+QB≤-2I4 (7)
wherein I4Representing an identity matrix of dimension 4, then introduces the following state transformations:
let z be [ z ]0,z1,z2,z3]TThe systems (2) and (3) can be converted into the following forms:
wherein B is0As defined above in the foregoing description,
constructing Lyapunov function V ═ z for nonlinear liquid level control resonant circuit systemTQz, the derivation of which can result in:
wherein c is 8Q, r is selected to be more than or equal to r*Max {1,4| | Q | } + δ, δ is an arbitrary normal number, so by selecting an appropriate constant gain r, it can be known from (10) that all states of the closed-loop system are asymptotically stable.
Preferably, the selected Hurwitz coefficient k0＝5,k1＝5,k2＝6,k3When r is 10, the controller of the system is:
while selecting the initial state of the system as [ x ]0(0),x1(0),x2(0),x3(0)]＝[1,1,1,1]。
Preferably, the method for determining the constant gain L is as follows:
2, leading: the presence of positive definite matrices Q and P makes the following holds:
wherein I4And I5Representing identity matrices of dimension 4 and dimension 5 respectively,
wherein a isi＞0,i＝1,2,3,4,hjThe parameters > 0, j ═ 1,2 and 3 are consistent with those in the observer and controller, and in fact, they can be arbitrarily selected, but in order to satisfy the above inequality, the solution verification can be performed by using a Linear Matrix Inequality (LMI) kit in Matlab.
The following dynamic transformation is then selected:
the system (2) and the observer (4) can be converted into the following form:
wherein:
constructing a Lyapunov function for a nonlinear level-controlled resonant circuit system
V＝zTQz+(m+1)εTPε (15)
Wherein m | | | Qa | | non-woven phosphor2Based on the theorem 2, deriving the lyapunov function can obtain:
wherein, theta is a constant obtained in the process of simplifying Lyapunov function, and the gain L is more than or equal to L by selecting a proper constant*Max {1, Θ (m +1) } + δ, δ is an arbitrary normal number, and all states of the resulting closed-loop system are asymptotically stable.
Preferably, the observer parameter a is selected1＝2,a2＝8,a3＝6,a410, L20 and a suitable controller parameter h1＝1,h2＝3,h3At this time, the form of control is as follows:
x1(0)＝1,x2(0)＝1,x3(0)＝1,
is the initial state of the selected system.
Compared with the prior art, the invention has the following effects: the invention considers the condition of input matching uncertainty, and respectively designs a state feedback controller and an output feedback controller based on a dynamic feedback control method. In the state feedback, the input matching uncertainty is introduced into the initial value of a newly added state variable by introducing integral control, so that direct processing is avoided, an extended low-gain observer with additional matching uncertainty estimation is designed in the output feedback, and finally, the extended low-gain observer and the extended low-gain observer can realize the asymptotic stability of the system.
Drawings
FIG. 1 is a schematic circuit diagram of a simple nonlinear liquid level control resonant circuit system;
FIG. 2 is a state feedback control neutral stateState x0A trajectory diagram of (a);
FIG. 3 shows a state x in the state feedback control1A trajectory diagram of (a);
FIG. 4 shows a state x in the state feedback control2A trajectory diagram of (a);
FIG. 5 shows a state x in the state feedback control3A trajectory diagram of (a);
FIG. 6 is a trace diagram of state w in state feedback control;
FIG. 7 shows state x in output feedback control1And
a trajectory diagram of (a);
FIG. 8 shows state x in output feedback control2And
a trajectory diagram of (a);
FIG. 9 shows state x in output feedback control3And
a trajectory diagram of (a);
FIG. 10 shows a state in output feedback control
A trajectory diagram of (a);
fig. 11 is a diagram of a locus of a state w in the output feedback control.
Detailed Description
A control method of a nonlinear liquid level control resonance circuit system based on dynamic feedback specifically comprises the following steps:
step 1: establishing a dynamic model of the nonlinear liquid level control resonant circuit system, selecting a proper state to establish state space expression:
as shown in fig. 1, in the modeling process of step 1, without considering the influence of deformation of some circuit elements in the nonlinear liquid level control resonant circuit system on the system model, the following dynamic description equation of the nonlinear liquid level control resonant circuit system can be obtained:
wherein u represents the control input voltage, VCRepresenting the voltage across the capacitor C, the resistances being R1,R2Inductors are respectively L1,L2The current through the inductor is iL1,iL2The current through the tunnel diode is id＝0.5sin(VC)。
Is to iL1The derivation is carried out by the derivation,
is to VCThe derivation is carried out by the derivation,
is to iL2And (6) derivation.
The system parameters are as follows:
table 2 defines x simultaneously1＝iL1,x2＝-VC,x3＝-0.5(iL2-0.5sin(VC) System (1) is transformed into the following state space representation form:
wherein w is 0.5iL2+0.125(iL2-0.5sin(VC))cos(VC) 0.5u-1 represents the control input, w *1 represents the input match uncertainty, which is considered due to the fact that: in some cases, the control value associated with the desired balance point may not be known, for example, in the case of set point adjustment or in the presence of sensor disturbances. And then, based on a dynamic feedback control method, respectively designing a state feedback controller and an output feedback controller for the system (2) to realize the gradual stabilization of the closed-loop system.
Step 2: designing a state feedback controller:
the state feedback controller is designed in the following form:
in the formula, ki> 0, i ═ 0,1,2,3, and are Hurwitz polynomials s4+k0s3+k1s2+k2s+k3R is a constant gain;
selected Hurwitz coefficient k0＝5,k1＝5,k2＝6,k3When r is 10, the controller of the system is:
while selecting the initial state of the system as [ x ]0(0),x1(0),x2(0),x3(0)]＝[1,1,1,1]As shown in fig. 2,3,4, and 5, the state x in the state feedback control is set to be the state x0、x1、x2、x3A track graph; as shown in fig. 6, which is a trace diagram of state w in the state feedback control;
and step 3: design of output feedback controller:
the system (2) sets the output y as x1＝iL2And only the output is measurable, the input and output information of the original system is utilized to reconstruct the system state, and the input matching uncertainty is subjected to self-adaptive estimation, so that the influence of the input matching uncertainty is eliminated. The observer was designed as follows:
the control input w is defined as follows:
wherein a isi> 0, i ═ 1,2,3,4, and are Hurwitz polynomials s4+a1s3+a2s2+a3s+a4Coefficient of (d), same hi> 0, i ═ 1,2,3, and is a Hurwitz polynomial s3+h1s2+h2s+h3L is a constant gain.
Selecting observer parameters a1＝2,a2＝8,a3＝6,a410, L20 and a suitable controller parameter h1＝1,h2＝3,h3At this time, the form of control is as follows:
x1(0)＝1,x2(0)＝1,x3(0)＝1,
is the initial state of the selected system; as shown in fig. 7, 8, 9, and 10, the state x in the feedback control is output for the state in the state feedback control1And
x2and
x3and
a trajectory diagram of (a); as shown in fig. 11, a trajectory diagram of state w in the output feedback control is shown.
Claims (5)
1. The control method of the nonlinear liquid level control resonance circuit system based on dynamic feedback is characterized by comprising the following steps:
step 1: establishing a dynamic model of the nonlinear liquid level control resonant circuit system, selecting a proper state to establish a state space expression, and obtaining a dynamic description equation of the nonlinear liquid level control resonant circuit system as follows:
wherein u represents the control input voltage, VCRepresenting the voltage across the capacitor C, the resistances being R1,R2Inductors are respectively L1,L2The current through the inductor is iL1,iL2，
Is to iL1The derivation is carried out by the derivation,
is to VCThe derivation is carried out by the derivation,
is to iL2Derivation is carried out;
defining x simultaneously1＝iL1,x2＝-VC,x3＝-0.5(iL2-0.5sin(VC) System (1) is transformed into the following state space representation form:
wherein w is 0.5iL2+0.125(iL2-0.5sin(VC))cos(VC) 0.5u-1 represents the control input, w*1 represents the input match uncertainty;
step 2: designing a state feedback controller:
the state feedback controller is designed in the following form:
in the formula, ki> 0, i ═ 0,1,2,3, and are Hurwitz polynomials s4+k0s3+k1s2+k2s+k3R is a constant gain;
and step 3: design of output feedback controller:
the system (2) sets the output y as x1＝iL2Only the output is measurable, the system state is reconstructed by using the input and output information of the original system, and the input matching uncertainty is subjected to self-adaptive estimation, so that the influence of the input matching uncertainty is eliminated; the observer was designed as follows:
the control input w is defined as follows:
wherein a isi> 0, i ═ 1,2,3,4, and are Hurwitz polynomials s4+a1s3+a2s2+a3s+a4Coefficient of (d), same hi> 0, i ═ 1,2,3, and is a Hurwitz polynomial s3+h1s2+h2s+h3L is a constant gain.
2. The control method of a nonlinear liquid level control resonant circuit system based on dynamic feedback as claimed in claim 1, wherein: the method for determining the constant gain r specifically comprises the following steps:
introduction 1: matrix arrayB0
Because B0Is a Hurwitz matrix, so that there is a suitable ki> 0, i ═ 0,1,2,3 and positive definite matrix Q ═ QTSatisfies the following conditions:
B0 TQ+QB≤-2I4 (7)
wherein I4Representing an identity matrix of dimension 4, then introduces the following state transformations:
let z be [ z ]0,z1,z2,z3]TThe systems (2) and (3) can be converted into the following forms:
wherein B is0As defined above in the foregoing description,
constructing Lyapunov function V ═ z for nonlinear liquid level control resonant circuit systemTQz, the derivation of which can result in:
wherein c is 8Q, r is selected to be more than or equal to r*Max {1,4| | Q | } + δ, δ is an arbitrary normal number, so by selecting an appropriate constant gain r, it can be known from (10) that all states of the closed-loop system are asymptotically stable.
3. The control method of a nonlinear liquid level control resonant circuit system based on dynamic feedback as claimed in claim 1, wherein:
selected Hurwitz coefficient k0＝5,k1＝5,k2＝6,k3When r is 10, the controller of the system is:
while selecting the initial state of the system as [ x ]0(0),x1(0),x2(0),x3(0)]＝[1,1,1,1]。
4. The control method of a nonlinear liquid level control resonant circuit system based on dynamic feedback as claimed in claim 1, wherein: the method for determining the constant gain L specifically comprises the following steps:
2, leading: the presence of positive definite matrices Q and P makes the following holds:
wherein I4And I5Representing identity matrices of dimension 4 and dimension 5 respectively,
wherein a isi＞0,i＝1,2,3,4,hjThe parameters are consistent with the parameters in the observer and the controller, and can be selected at will in fact, but the Linear Matrix Inequality (LMI) tool box in Matlab can be used for solving and verifying to meet the inequality;
the following dynamic transformation is then selected:
the system (2) and the observer (4) can be converted into the following form:
wherein:
constructing a Lyapunov function for a nonlinear level-controlled resonant circuit system
V＝zTQz+(m+1)εTPε (15)
Wherein m | | | Qa | | non-woven phosphor2Based on the theorem 2, deriving the lyapunov function can obtain:
wherein, theta is a constant obtained in the process of simplifying Lyapunov function, and the gain L is more than or equal to L by selecting a proper constant*Max {1, Θ (m +1) } + δ, δ is an arbitrary normal number, and all states of the resulting closed-loop system are asymptotically stable.
5. The control method of a nonlinear liquid level control resonant circuit system based on dynamic feedback as claimed in claim 1, wherein:
selecting observer parameters a1＝2,a2＝8,a3＝6,a410, L20 and a suitable controller parameter h1＝1,h2＝3,h3At this time, the form of control is as follows:
x1(0)＝1,x2(0)＝1,x3(0)＝1,
is the initial state of the selected system.
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