CN107272416A - One class Linear Parameter-Varying Systems dynamic quantization H ∞ control methods - Google Patents
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Abstract
The present invention discloses a class Linear Parameter-Varying Systems dynamic quantization H ∞ control methods, consider that linear dimensions variation control system has bounded non-linear, in the case of quantization and model parameter time-varying, initially set up closed loop linear dimensions variable quantity networked control systems model, reconstruct appropriate Lyapunov functions, utilize Lyapunov Theory of Stability and LMI analysis method, obtain stable and the presence of H ∞ controllers the adequate condition of system, simultaneously using many cell space technologies and quantified controlling strategy by Solve problems that the Solve problems of infinite dimensional LMI group are approximately finite dimensional linear MATRIX INEQUALITIES group.Finally solved using Matlab LMI tool boxes, the dynamic quantization H ∞ controller gains matrix for obtaining Linear Parameter-Varying Systems is K (ρ (k))=W (ρ (k)) X‑1.The present invention considers dynamic quantization and random perturbation situation, is more of practical significance, it is adaptable to general H ∞ controls, reduces the conservative of H ∞ controller designs.
Description
Technical Field
The invention relates to control of a networked system, in particular to a dynamic quantization H-infinity control method of a linear parameter change system.
Background
In recent years, gain scheduling control, especially based on Linear Parameter-Varying (LPV) system, has been increasingly studied. LPV systems are a class of systems in which the parameters are constantly changing, the elements of the state matrix of such systems being deterministic functions with time-varying parameters, and the range of time-varying parameters associated with the functions being measurable. Many practical systems can be described by the above models, such as aircraft systems, wind energy conversion systems, etc., and many results and research reports have been made on the research of such systems. Due to the fact that the nonlinear and time-varying characteristics of a type of actual dynamic system can be described, a linear parameter varying system becomes a hot spot which is paid much attention by the control theory in recent years, and the theory of the linear parameter varying system is successfully applied to the fields of aviation, aerospace, robots, industrial process control and the like. Meanwhile, in an actual feedback control system, considering the limitation of network transmission capacity or bandwidth, a quantizer is required to process transmission signals in a data channel, and these quantization characteristics will often seriously affect the stability of the system and deteriorate the performance index of the system, so that it is necessary to research the quantization control of a linear parameter variation system.
The rapid development of modern network control systems has its enormous advantages, such as low cost, simple installation and maintenance, high reliability and flexibility, strong fault tolerance and fault diagnosis capability, and convenient remote operation and control. However, when the LPV system is introduced into a network, new problems, such as data quantization, network delay, data packet loss, etc., occur, and the performance of the entire system is reduced, or even the entire system is unstable. The existence of quantization makes the analysis and synthesis of the system more complicated and difficult, and the quantization error caused by the existence of quantization is also one of the sources of system instability and system performance deterioration. Therefore, the method has important theoretical significance and wide application prospect for the dynamic quantization H-infinity control research of the linear parameter change system.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a dynamic quantization H-infinity control method of a linear parameter change system. Under the condition that quantization and controller gain change exist in the linear parameter change control system, the linear parameter change system dynamic quantization H-infinity controller is designed, so that the linear parameter change quantization control system can still keep system stability under the condition and meet performance indexes.
The technical scheme adopted by the invention is as follows: a dynamic quantization H-infinity control method of a linear parameter variation system comprises the following steps:
1) a dynamic quantization state feedback controller is designed for a linear parameter change system, and the model of the linear parameter change system is as follows:
wherein:
is a state variable of the system;is the control output of the system;is a control input to the system;
for the disturbance input of the system, { Φ (k), k ═ d, -d +1, …,0}, is a known initial sequence of conditions αiCoordinates of multicellular bodies, αi≥0，Ai,B1i,B2i,Ci,DiSystem matrices a (ρ (k)), B of the LPV system, respectively1(ρ(k))，B2(ρ (k)), C (ρ (k)), and D (ρ (k)) values at each vertex of the multicellular body, with the parameter vector ρ (k) ═ ρ1(k)ρ2(k)…ρs(k)]TSatisfy rhoi(k) Is measurable in real time, and iρis ρi(k) The lower limit of (a) is,is ρi(k) The upper limit of (3).
The dynamic quantizer is:
and, the quantizer satisfies:
wherein μ > 0 is a quantization parameter,is a quantized variable, qμ(z) represents the amount after dynamic quantization, and M and Δ represent the quantization range and quantization error, respectively. The quantization range of the dynamic quantizer is μ M, and the quantization error is μ Δ. μ can be viewed as a scaling variable: amplifying mu to obtain a quantizer with larger quantization error and quantization range, so that any large signal can be measured; reducing mu can obtain a quantizer with smaller quantization error and quantization range, so that any small signal can be measured, and the signal is further driven to the origin.
With the quantizer of equation (2), for system (1), the dynamic quantization feedback controller with two quantized signals is designed as follows:
wherein, mu1Is a dynamic parameter, μ, of the quantizer 12Is a dynamic parameter of the quantizer 2 and,andis a dynamic quantizer obtained according to equation (2). QuantizerHas a quantization range of μ1M1Quantization error of μ1Δ1QuantizerHas a quantization range of mu2M2Quantization error of μ2Δ2K (ρ (K)) is a systemA gain matrix of the system.
In the designed quantization control strategy, the following relation is required to be satisfied:
where θ > 0 is a tunable constant.
2) Obtaining a controller meeting H-infinity performance index;
first, construct the Lyapunov function:
V(k)＝x(k)TPx(k)，P＝X-1
then sufficient conditions exist for the H ∞ controller of the system to stabilize and dynamically quantify the system for linear parameter changes to be: given an index of γ > 0, assume the presence of the matrix WiI ═ 1,2, … r and the positive definite symmetric matrix X > 0, such that the following linear matrix inequality holds:
wherein: represents a transpose of the symmetric position matrix;
if true, one can obtain:
Ki＝WiX-1(7)
is a symmetrical positive definite matrix and is characterized in that,is a symmetric positive definite matrix, KiA controller gain matrix at the vertex for the system; gamma is given normal number, using Matlab LMI toolbox to solve ifThere is a symmetric positive definite matrix Real number gamma>0, then the H-infinity control system of the linear parameter varying system is stable and satisfies the H-infinity performance index, and the gain matrix of the controller at each vertex is Ki＝WiX-1The whole controller gain matrix is
According to γ ═ Σ (| | z)k||)/Σ(||wkI) to obtain the optimal disturbance rejection ratio gamma under the control of the corresponding system performance index gamma, H infinityoptThe optimized conditions are as follows:
if the solution exists, the optimal disturbance rejection ratio gamma of the closed-loop system under the condition of conforming to the H-infinity control condition of the linear parameter variation system can be obtainedoptWhile linear parameter change quantifies the system controller gain matrix KiIs optimized to K* i＝WiX-1The whole controller gain matrix is
3) Designing a quantitative control strategy to obtain the quantitative error of the system;
if equation (6) holds, then there is a symmetric positive definite matrix P > 0 and
such that the following inequality holds:
wherein,is a symmetric positive definite matrix, a matrixRepresents a transpose of the symmetric position matrix;
consider a system (1) controlled by a quantization controller (4) if M1The selection is sufficiently large and meets the following requirements:
wherein, then λmin(Q) is the minimum eigenvalue of matrix Q; get
Finally, μ is obtained from formula (5)2＝θμ1Then, the controller u (k) of the system is obtained by the equation (4), so that the dynamic quantization H ∞ control of the linear parameter variation system can be realized.
The invention designs feedback gains meeting H infinity performance and dynamic characteristics for each vertex of a multicellular LPV system by utilizing a gain scheduling technology, an H infinity theory and an LPV convex decomposition technology, firstly obtains a gain matrix of each vertex of the multicellular LPV system, and finally obtains the LPV controller of the system comprehensively. Secondly, considering the quantization error, under the designed quantization control strategy, the closed-loop LPV system is asymptotically stable and has a specified H ∞ performance index.
Compared with the prior art, the invention has the following beneficial technical effects:
1) aiming at a linear parameter change control system, the invention simultaneously considers incomplete measurement factors and uncertain factors in the system, including quantization, external disturbance and uncertainty caused by model simplification or other physical factors existing in the system, establishes a closed-loop linear parameter change control system model through a series of derivation and conversion, and provides a design method of a quantization H-infinity controller of the system;
2) the invention considers the quantization factor of signal transmission, and has more practical significance;
3) the invention is suitable for general H-infinity control, and reduces the conservatism of the design method of the linear parameter change controller;
4) the dynamic quantizer adopted by the invention has unique advantages compared with a static quantizer, and can adjust dynamic parameters in real time according to quantization errors, thereby reducing the influence on the whole system.
Drawings
FIG. 1 is a flow chart of a linear parameter variation system dynamic quantization H ∞ control method.
Fig. 2 shows γ -5.9799, ω -0.5 sin (k), and M is taken as the quantization range1＝69，M2The linear parameter variation system at 28.76 dynamically quantizes the H ∞ open loop control state response map.
Fig. 3 shows γ -5.9799, ω -0.5 sin (k), and M is the quantization range1＝69，M2The linear parameter change system at 28.76 dynamically quantizes the H ∞ closed-loop control state response map.
Fig. 4 shows γ -5.9799, ω -0.5 sin (k), and M is the quantization range1＝69，M2The linear parameter change system dynamically quantizes the state response diagram before and after H-infinity closed-loop control quantization at 28.76.
Fig. 5 shows γ -5.9799, ω -0.5 sin (k), and M is the quantization range1＝69，M2And (3) dynamically quantizing the adjusted response diagram of the H-infinity closed-loop control system by the linear parameter change system at 28.76.
Detailed Description
The following further describes the embodiments of the present invention with reference to the drawings.
Referring to fig. 1, a dynamic quantization H ∞ control method for a linear parameter variation system includes the following steps:
step 1: a dynamic quantitative feedback controller is designed for a linear parameter change system, and a linear parameter change system model is an expression (1). The dynamic quantizer is equation (2), and the quantizer satisfies equation (3).
With the quantizer defined by equation (2), for system (1), a dynamic quantization feedback controller having two quantized signals is designed by equation (4), and equation (5) is required to be satisfied in the designed control strategy.
By combining the LPV system and the dynamic quantization controller, a quantization closed-loop control system can be obtained as shown in formula (13):
wherein,
step 2: obtaining a controller meeting H-infinity performance index;
first, construct the Lyapunov function:
V(k)＝x(k)TPx(k)，P＝X-1
wherein P > 0 and satisfies
The inequality (14) holds according to Schur's theorem
Wherein,denotes the transpose of the symmetric position matrix.
According to the Lyapunov stability theory, the sufficient condition for the linear parameter variation quantization control system shown in the formula (14) to be stable is as follows: presence of positive definite matrixγ > 0, such that the linear matrix inequality (15) holds. The condition (15) is subjected to matrix transformation, thereby obtaining expression (6).
Based on a Lyapunov function, obtaining sufficient conditions for stabilizing and quantizing an H-infinity controller by a linear parameter change control system by utilizing a Lyapunov stability theory and a linear matrix inequality analysis method: given an index of γ > 0, assume the presence of the matrix WiWhen i is 1,2, … r and the positive definite symmetric matrix X > 0, equation (6) is satisfied, and equation (7) is obtained.
According to γ ═ Σ (| | z)k||)/Σ(||wkI) to obtain the optimal disturbance rejection ratio gamma under the control of the corresponding system performance index gamma, H infinityoptIf the optimization condition is that the formula (8) is satisfied, the optimal disturbance rejection ratio gamma of the system can be obtained under the condition that the closed-loop system conforms to the H-infinity control condition of the linear parameter variation systemoptWhile linear parameter change quantifies the system controller gain matrix KiIs optimized to K* i＝WiX-1The whole controller gain matrix is
And step 3: designing a quantization control strategy of the system to obtain a quantization error of the system;
if equation (6) holds, then there are symmetric positive definite matrices P > 0 and equation (9) such that equation (10) holds.
Consider a system (1) controlled by a quantization controller (4) if M1The selection is sufficiently large and satisfies the following equations (11) and (12).
Thus, μ is obtained from formula (5)2＝θμ1Then, the controller u (k) of the system is obtained by the equation (4), so that the dynamic quantization H ∞ control of the linear parameter variation system can be realized.
Example (b):
the invention provides a dynamic quantization H-infinity control method of a linear parameter change system. The specific implementation method comprises the following steps: step 1: the controlled object is a linear parameter change quantitative control system, the state space model of the controlled object is expressed by formula (1), and the system parameters are given as follows:
where ρ is1(k)，ρ2(k) Is a time-varying parameter which can be measured in real time and has a value range of rho1(k)∈[0.5,1.5]，ρ2(k)∈[1,2]Then, the vertex of the parameter polyhedron Θ is:let θ be 2, quantize error Δ1＝Δ2Using formula (12), Q (ξ) > 0, γ 5.9799 and ξ are 0.1, and further calculated as:
the quantization range of the quantizer may be taken as M1When the formula (5) is repeated, M can be obtained as 692＝28.76。
To further illustrate the effectiveness of the design method, values calculated above are used to take α1＝α2＝α3＝α4＝1/4。
The disturbances are:
step 2: and (3) selecting different values aiming at rho (k), giving different values according to the step 1, and solving the parameters of the controller under the conditions of different vertexes and corresponding H-infinity performance indexes through a Matlab LMI tool box.
Assuming H ∞ control: solving parameters and H infinity disturbance suppression level gamma after controller optimization under different conditions through Matlab LMI toolboxoptAnd using gamma-sigma (| | z)k||)/Σ(||wk| l) to find the corresponding system performance index γ. As the parameters in the system change, the gain of the controller of the system also changes, which shows that the parameters have important influence on the performance of the system. And, gamma < gammaoptThe same holds for different parameters, indicating that the designed controller meets the H ∞ performance index.
And step 3: giving an initial state of x (0) ═ 1-1]TAnd (3) simulating the state response of the system under the H-infinity control under different conditions by using the result of the solution of the LMI toolbox in the Matlab in the step 2 by using the Matlab, as shown in the attached figures 2 to 5, wherein the attached figure 2 shows that the system is divergent and unstable in the open loop state. As can be seen from fig. 3, the closed loop system formed by the controller designed by the present invention is asymptotically stable, wherein a disturbance is given to the system at the time when k is 10, as shown in fig. 3, the system state fluctuates, and finally, the system state tends to be a stable state, which illustrates the effectiveness of the designed controller. And figure 4 is the state x of the system1In the graphs before and after quantization, it can be observed from the graphs that due to the introduction of the quantizer, the generated quantization error can cause errors in the state curves before and after quantization, and the quantization curve more intuitively illustrates that the quantization error can deteriorate the stability or performance index of the system. From the formula (10), it can be analyzed that the optimum lambda needs to be found using the LMI toolboxmin(Q), and further an optimal quantization range M is obtained1Therefore, the error of the whole quantized LPV system is reduced to the minimum, the state curves of the system before and after quantization, which are formed by the figure 4, are infinitely close, and the quantized control strategy is more complete. Wherein figure 5 is a tuned output response curve of the system. As is clear from the actual graph, when a disturbance occurs, the output of the system is suppressed to the suppression ratio of the disturbance This shows that the controller designed by the invention can ensureThe closed loop system is proved to be asymptotically stable and meet the given performance index.
The present invention is not intended to be limited to the particular embodiments shown above, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
Claims (1)
1. A linear parameter change system dynamic quantization H infinity control method is characterized by comprising the following steps:
1) a dynamic quantitative feedback controller is designed for a linear parameter change system, and the model of the linear parameter change system is as follows:
x(k)＝Φ(k),k∈[-d,0]
wherein:
is a state variable of the system and is,is the control output of the system and is,is a control input to the system and is,for the disturbance input of the system, { Φ (k), k ═ d, -d +1, …,0}, a known initial sequence of conditions, αiIs the coordinate of a multicellular body, and, αi≥0，Ai,B1i,B2i,Ci,DiSystem matrices a (ρ (k)), B of the linear parameter varying system, respectively1(ρ(k))，B2(ρ (k)), C (ρ (k)), and D (ρ (k)) value parameter vectors at each vertex of the multicellular body, ρ (k) ═ ρ1(k) ρ2(k) … ρs(k)]TSatisfy rhoi(k) The range can be measured, an iρIs ρi(k) The lower limit of (a) is,is ρi(k) The upper limit of (d);
the dynamic quantizer is:
and, the quantizer satisfies:
wherein: μ > 0 is a quantization parameter which,is a quantized variable, qμ(z) represents the amount after dynamic quantization, M and Δ represent the quantization range and quantization error, respectively, the quantization range of the dynamic quantizer is μ M, and the quantization error is μ Δ;
with the quantizer of equation (2), for system (1), the dynamic quantization feedback controller with two quantized signals is designed as follows:
wherein: mu.s1Is a dynamic parameter, μ, of the quantizer 12Is a dynamic parameter of the quantizer 2 and,andis a dynamic quantizer obtained according to equation (2); quantizerHas a quantization range of μ1M1Quantization error of μ1Δ1QuantizerHas a quantization range of μ2M2Quantization error of μ2Δ2K (ρ (K)) is the gain matrix of the system;
in the designed quantization strategy, the following relation is required to be satisfied:
wherein: θ > 0 is a tunable constant;
2) the method comprises the following steps of obtaining a controller which meets an H-infinity performance index:
first, construct the Lyapunov function:
V(k)＝x(k)TPx(k)，P＝X-1
then sufficient conditions exist for the H ∞ controller of the system stability and linear parameter variation quantization control system to be: given an index of γ > 0, assume the presence of the matrix WiI ═ 1,2, … r and the positive definite symmetric matrix X > 0, such that the following linear matrix inequality holds:
wherein: represents a transpose of the symmetric position matrix;
if true, one can obtain:
Ki＝WiX-1(7)
is a symmetrical positive definite matrix and is characterized in that,is a symmetric positive definite matrix, KiA controller gain matrix at the vertex for the system; gamma is given normal number, and the solution is carried out by utilizing Matlab LMI tool box if a symmetric positive definite matrix exists Real number gamma>0, then the H-infinity control system of the linear parameter varying system is stable and satisfies the H-infinity performance index, and the gain matrix of the controller at each vertex is Ki＝WiX-1The whole controller gain matrix is
According to γ ═ Σ (| | z)k||)/Σ(||wkI) to obtain the optimal disturbance rejection ratio gamma under the control of the corresponding system performance index gamma, H infinityoptThe optimized conditions are as follows:
if the solution exists, the optimal disturbance rejection ratio gamma of the closed-loop linear parameter variation system can be obtained under the condition of meeting the H-infinity controloptWhile linear parameter change quantifies the system controller gain matrix KiIs optimized to K* i＝WiX-1The whole controller gain matrix is
3) Designing a quantitative control strategy to obtain the quantization error of the system:
if equation (6) holds, then there is a symmetric positive definite matrix P > 0 and
such that the following inequality holds:
wherein,is a symmetric positive definite matrix, a matrixRepresents a transpose of the symmetric position matrix;
consider a system (1) controlled by a quantization controller (4) if M1The selection is sufficiently large and meets the following requirements:
wherein, Delta is theta Delta2+||K(ρ(k))||Δ1, Then λmin(Q) is the minimum eigenvalue of matrix Q; get
Finally, μ is obtained from formula (5)2＝θμ1Then, the controller u (k) of the system is obtained by the equation (4), so that the dynamic quantization H ∞ control of the linear parameter variation system can be realized.
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| CN109150639B (en) * | 2018-11-05 | 2021-07-09 | 江南大学 | Finite time domain H-infinity control method of time-varying system under influence of high-rate communication network |
| CN110262334A (en) * | 2019-06-17 | 2019-09-20 | 江南大学 | The finite time-domain H ∞ control method of state saturation system under a kind of random communication protocol |
| CN110262334B (en) * | 2019-06-17 | 2021-05-28 | 江南大学 | Finite time domain H-infinity control method of state saturation system under random communication protocol |
| CN113917842A (en) * | 2021-10-14 | 2022-01-11 | 北京理工大学重庆创新中心 | Data driving control method of networked control system and controller thereof |
| CN113917842B (en) * | 2021-10-14 | 2024-07-26 | 北京理工大学重庆创新中心 | Data driving control method of networked control system and controller thereof |
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