A Design Method of Automotive Suspension Sampling Controller Based on Discrete System

Abstract

The present invention discloses a non-periodic sampling data H-infinity control method for an automotive active suspension system based on an uncertain discrete-time model. The present invention relates to the non-periodic sampling data H-infinity control problem of an uncertain discrete-time model and an automotive active suspension system with time-varying time delay. The steps of the present invention are as follows: Step 1: Discretize the system by using a lifting technique, and quantify the uncertainties caused by sampling data and delay by finding the integral quadratic constraint multiplier; Step 2: Through model transformation, give the conditions for the exponential stability of the closed-loop system, and use the KYP lemma to transform it into an LMI condition; Step 3: Utilize the energy function of the uncertainty operator and the integral quadratic constraint characteristics to optimize the H-infinity performance measurement ride comfort; Step 4: Design an algorithm for obtaining the controller gain; Step 5: Take a four-section vehicle model as an example to verify the effectiveness and optimization effect of the method. The present invention is used in the field of automotive active suspension control.

Description

Technical Field

[0001] The present invention relates to the field of automotive active suspension control, and more particularly to a non-periodic sampling data H-infinity control method for an automotive active suspension system based on an uncertain discrete-time model. Background Art

[0002] An automotive suspension is an important device for transmitting and filtering forces between the road surface and the vehicle body, and plays an important role in improving vehicle performance. Generally speaking, the main performance requirements include: eliminating the rocking impact on passengers caused by rough roads (riding comfort); ensuring firm holding and uninterrupted contact of the wheels with the road (driving safety), and restricting the suspension stroke and hydraulic actuators. However, these requirements are conflicting and it is difficult to optimize them simultaneously. As a control method for balancing these performance requirements, active suspension has received increasing attention in recent years.

[0003] Currently, in order to improve the performance of active suspension systems, many control methods have been developed, such as finite-time control, fuzzy control, event-triggered control, adaptive control, etc. Among numerous studies, due to the H-infinity performance index establishing a constraint relationship between the disturbance input and the system output, H-infinity control has become a natural choice for optimizing the performance of active suspension systems. A robust controller designed based on the H-infinity control theory takes into account the influence of more general road surface disturbances. In addition, using the H-infinity performance to measure riding comfort and specifically constraining different performance requirements avoids the conservatism brought about by constraining all requirements in a single objective function. The research on frequency-band H-infinity performance constraints effectively overcomes the problem of over-constraint in the full frequency domain. Thus, how to more optimally establish performance constraints and more accurately estimate performance under system uncertainties is a topic worthy of discussion.

[0004] With the development of computers and digital controllers, sampled-data control has received increasing attention. There are many mature studies on sampled-data control, such as the input delay method; the loop-based functional method; the switched system method and the stochastic system method, and some of these methods have inspired the exploration of active suspension performance. The input delay method (journal: IEEE Transactions on Control Systems Technology; authors: Huijun Gao, Weichao Sun and Peng Shi; publication time: 2010; article title: Robust Sampled-Data H∞ Control for Vehicle Active Suspension Systems; pages: 238-245) is used to design the H-infinity controller of the aperiodic sampled-data active suspension system; the input delay method (journal: IEEE / CAA Journal of Automatica Sinica; authors: Wenfeng Li, Zhengchao Xie, Yucong Cao, Pak Kin Wong and Jing Zhao; publication time: 2021; article title: Sampled-Data Asynchronous Fuzzy Output Feedback Control for Active Suspension Systems in Restricted Frequency Domain; pages: 1052-1066) is used for model conversion to analyze the fuzzy sampled-data control of the active suspension system; the loop-based functional method (journal: International Conference on Control, Decision and Information Technologies (CoDIT); authors: Seungyong Han, S.M. Lee, Ho-Youl Jung and Ju H. Park; publication time: 2019; article title: Constrained H∞ Control for Active Suspension Systems with Aperiodic Sampling: a Looped Functional Approach; pages: 796-800) is used to design the sampled-data H-infinity controller of the active suspension system. These studies are all carried out under the continuous-time model based on Lyapunov theory, although the controllers are digital. In the continuous-time model, the piecewise continuity of the sawtooth function representing the time span during which the signal is held by the zero-order hold is approximately regarded as continuous, which leads to conservatism. However, there are few studies on the sampled-data H-infinity control under the discrete model of the suspension system.On the one hand, it is necessary to transform it into an equivalent discrete-time model of the active suspension, and this process is affected by the sampling data interval and time delay. On the other hand, under the Lyapunov theory, difference equations are more difficult to handle than differential equations.

[0005] To address the above problems, this paper studies the discrete aperiodic sampled-data H-infinity control problem of an active suspension system by means of the analysis of uncertain operators. After model transformation, the uncertain discrete-time model is transformed into a feedback interconnection composed of a linear time-invariant system and a time-varying operator. Combining the constraint conditions of the operator, the exponential stability condition and the H-infinity performance constraint are obtained. Further, based on these conditions, a control algorithm is given. During this process, we note that the use of the integral quadratic constraint (IQC) theory requires the initial state to be zero, which will limit the analysis of the active suspension system. To solve this problem, the Bohl-Perron principle is used to transform the system with an arbitrary initial state into an equivalent system with a zero initial state, and then the exponential stability of the original system is transformed into the l2 stability of the new feedback interconnection, which has higher practical application prospects. Summary of the Invention

[0006] The object of the present invention is to solve the discrete aperiodic sampled-data H-infinity control problem of an active suspension system, and to propose an aperiodic sampled-data H-infinity control method for an automotive active suspension system based on an uncertain discrete-time model.

[0007] An aperiodic sampled-data H-infinity control method for an automotive active suspension system based on an uncertain discrete-time model includes the following steps:

[0008] Step 1: Establish an uncertain discrete-time model;

[0009] Step 2: Give the performance constraint conditions and stability conditions of the suspension system;

[0010] Step 3: Give the construction process of the equivalent feedback interconnection;

[0011] Step 4: Give the algorithm for solving the controller;

[0012] As a further improvement of the present invention, the specific process of establishing the uncertain discrete-time model in Step 1 is as follows:

[0013] Consider an active vehicle suspension system with time-varying delay;

[0014] According to Newton's second law, the dynamic equations of the sprung mass and the unsprung mass are described as:

[0015]

[0016] By defining the following state variable, x1(t) = ys (t) - y u (t), x2(t) = y u (t) - y r (t), The disturbance input is The state x(t) = col{x1(t), x2(t), x3(t), x4(t)}, and the dynamic equation (1) is rewritten as the following state - space model:

[0017]

[0018] where,

[0019] B u = [0 0 1 / m s - 1 / m u T , B w = [0 - 1 0 c t / m u T .

[0020] The specific process of giving the performance constraint conditions and stability conditions of the suspension system in the second step is as follows:

[0021] Consider the following performance requirements:

[0022] ① Ride comfort; As is well - known, ride comfort is related to the body acceleration and is used as the performance output of the state - space model (2), which can be denoted as z1(t).

[0023] ② Suspension stroke; Considering the mechanical structure, the suspension stroke should not exceed the maximum allowable value y max , that is, |y s (t) - y u (t)| ≤ y max ;

[0024] ③ Road - holding; For driving safety, the contact between the tire and the road surface should be firm and continuous, and the tire dynamic load should be small, expressed as k t (y u (t) - y r (t)) ≤ 9.8(m s + m u );

[0025] ④ Actuator power limit; Due to the high power consumption of the active suspension system, the power applied to the actuator by the active suspension control should also be limited, that is, |u(t)| ≤ u max ;

[0026] ​​Considering the above four constraints and reflecting them in the input and output of the system, the vehicle active suspension model is expressed as:

[0027]

[0028] Among them, C1 = [-k s / m s 0 -c s / m s c s / m s , D1 = 1 / m s ,

[0029] Due to the time-varying delay in the signal transmission process, the signal cannot be updated after sampling; in order to study the non-periodic sampling data control problem of the active suspension system, the following assumptions are made;

[0030] Assumption 1; Assume that {t k} is the sampling data sequence, satisfying Among them is the maximum sampling data interval;

[0031] Since the delay in the actual system is usually less than one sampling data interval, consider the case of τ k <h k According to the above description, the research control law is

[0032] for the design problem of state feedback control, where when k = 0, x t-1 = x init ;

[0033] On the sampling data interval, according to the characteristics of the input control law (4), solve the state equations at the sampling data time t k and the signal update time a k to obtain the discrete-time system model when x(t0) = x0:

[0034]

[0035] Among them,

[0036] As a further improvement of the present invention, the specific process of the construction process of the equivalent feedback interconnection given in the third step is:

[0037] In order to analyze the discrete-time model (5), let h k = h0 + θ k , where is a constant, is the time-varying part; through the following two propositions, it is transformed into an equivalent feedback interconnection composed of an LTI system and a time-varying operator;

[0038] Proposition 1; Assume h k = h0 + θ k , where h0 is a fixed constant. By lifting the state and deforming the model, the discretized system is represented as the following uncertain discrete-time model:

[0039]

[0040] where,

[0041] By lifting the state Combined with the operator Prove that the discrete-time system model (5) and the uncertain discrete-time model (6) are equivalent.

[0042] Proposition 2; The uncertain discrete-time model (6) is converted into the following interconnection model

[0043]

[0044] where,

[0045] By defining new inputs and outputs, all uncertainty operators are represented by diagonal operators, and it can be seen that the uncertain discrete-time model (6) and the interconnection model (7) are equivalent;

[0046] Through the above two propositions, the discrete-time system model (5) is transformed into the interconnection model (7), paving the way for the application of the IQC theorem in subsequent work; on the other hand, note that the conversion between the interconnection model (7) and the discrete-time system model (5) is reversible, so the analysis of the discrete-time system model (5) is transformed into the analysis of the interconnection model (7).

[0047] As a further improvement of the present invention, the specific process of the algorithm for solving the controller given in the fourth step is as follows:

[0048] Analyze the stability and H-infinity performance of the system based on the IQC method;

[0049] When w(t) = 0, the interconnection model (7) is written as the following feedback interconnection:

[0050]

[0051] Since the feedback interconnection (8) cannot guarantee the condition of zero initial state, the IQC method cannot be directly used; to solve this problem, there is the following theorem;

[0052] Theorem 1. The pseudo matrix satisfies Then the feedback interconnection (8) is exponentially stable if, for any Ψ(t k ) ∈ l1[0, ∞), the following interconnection is l2-stable. The interconnection with zero initial state:

[0053]

[0054] where, when k ≥ 1 and when k = 0 In the feedback interconnection (8) and are the same;

[0055] Δ1 and Δ2 are exponential functions of matrix A; to analyze the upper bound of the l2-norm of the uncertain operator, the following Schur decomposition of matrix A is performed:

[0056] U T AU = D + E, (10)

[0057] where U is a unitary matrix, D is a diagonal matrix, and E is a strictly upper triangular matrix. Based on this decomposition, the following results are obtained;

[0058] Lemma 1; for any and κ ∈ [0, 1], the operator obtained in the feedback interconnection (8) satisfies the following IQC as shown below:

[0059]

[0060] where,

[0061] R1 = diag{R 11 , R 12 , R 13 , R 14}, Υ1 = diag{γ1I, γ1I, γ2I, γ1γ2I}, and α1 = λ max (A), α2 = λ max (-A);

[0062] It is obtained that is correct; thus, for the operators Δ1 and Δ2, there exists ​Therefore, for any κ ∈ [0, 1],

[0063]

[0064] Therefore, inequality (11) holds for any κ ∈ [0, 1];

[0065] Corollary 1; For any The operator given in Proposition 2 satisfies the following inequality constraints:

[0066]

[0067] where Υ2 = diag{γ1I, γ1I, γ2I, γ1γ2I, γ1I}, R2 = diag{R 21 , R 22 , R 23 , R 24 , R 25}, ε′ > 0, i = 1, 2, 3, 4, 5;

[0068] Lemma 2; Given a positive scalar γ, the system (7) satisfies If there exists R2 = diag{R 21 , R 22 , R 23 , R 24 , R 25}, such that

[0069]

[0070] where, Θ1 = diag{P2, -P2}, Θ2 = diag{Υ2R2, I, -R2, -γ 2 I};

[0071] For the storage function V(x(t k )) = x(t k ) T P2x(t k ), multiplying both sides of F(P2, R1′) by col{x(t k ), u(t k ), w(t k )}T and col{x(t k ), u(t k ), w(t k )}, it is obtained that inequality (13) is equivalent to

[0072] V(x(t k+1)) - V(x(t k )) - γ 2 w T (t k )w(t k ) + z T (t k )z(t k ) + Υ2y T (t k )R2y(t k ) - u T (t k )R2u(t k ) < 0,

[0073] This means that

[0074]

[0075] Since V(x(t k )) > 0 and x0 = 0, it is known that and V(x0) = 0, and further according to Corollary 1, it is known that holds;

[0076] Theorem 2; Assume that Given a scalar γ > 0, and the state - feedback controller gain K exists in the form of control law (4). If there exist scalar matrices symmetric matrices and the following linear matrix inequalities are applicable:

[0077]

[0078] F(P2, R2) < 0, (16)

[0079]

[0080]

[0081] where S1 = [1 0], S2 = [0 1], N = diag(K, K), ρ = γ 2 w max + V(x(t0)).

[0082] First, prove that inequality (15) guarantees the exponential stability of the feedback interconnection (8); According to the KYP lemma, it is found that if inequality (15) holds, then the following inequality

[0083]

[0084] is established, where is the transfer matrix of G in the feedback interconnection (9). Thus, there exists ε > 0 such that

[0085]

[0086] is established for all Lemma 1 shows that for any κ ∈ [0, 1], satisfies the IQC defined by Π; since is linear and causal, the feedback interconnection is well-posed; thus, according to the IQC theorem, the transfer matrix (9) is l2-stable, and then according to Theorem 1, the feedback interconnection (8) is exponentially stable;

[0087] Secondly, according to Lemma 2, when the inequality (16) is satisfied, there is

[0088] Thirdly, define and ρ = γ 2 w max + V(x0). According to the proof of the input and output constraints, it is known that if the inequalities (17) and (18) hold, then |u(t k )| ≤ u max and |{z2(t k )} m | ≤ 1, m = 1, 2 hold.

[0089] Theorem; for the state-space model (2), assume Considering a scalar γ > 0, and a suitable ε > 0, if there exist a symmetric matrix M1 > 0, a scalar matrix and a matrix such that

[0090] W1 T Λ1W1 - εM1 < 0, (20)

[0091]

[0092]

[0093]

[0094] then the state feedback controller gain matrix K for the performance index is obtained, K = [1 0]N[1 0] T , where M1 = [N 0 -I] T [N 0 -I], M2 = [N 0 0 -I] T [N 0 0 -I], Λ1 = diag{P1, -P1, Υ1 2R1, -R1} ρ = γ 2 w max + V(x(t0));

[0095] Combining the definitions in Propositions 1 and 2, let

[0096]

[0097] Let and be decomposed into where

[0098]

[0099] For a suitable ε > 0, we get

[0100]

[0101]

[0102] From Lemma 1, we know that are respectively equal to Inequality (15) and Inequality (16); thus, the controller finds N such that Inequalities (20)-(23) hold, and K = [1 0]N[1 0] T ;

[0103] To solve the controller gain matrix, the following iterative algorithm is given:

[0104] S1: Give the upper bound interval of the sampling data and the upper bound interval of the delay An initial value γ > 0 and a scalar ε = 1;

[0105] S2: Let N T N = M, solve the LMIs (20)-(23) to obtain the controller gain matrix, where K = [1 0]N[1 0] T ;

[0106] S3: If there exist feasible solutions for the LMIs (15)-(18) and the K obtained above;

[0107] S4: γ = γ - 0.01, repeat Steps 2 - 3;

[0108] S5: else;

[0109] S6: Let ε = ε * 0.1, repeat Steps 2 - 5;

[0110] S7: endif;

[0111] S8: Output the controller gain matrix K and the H-infinity performance index γ;

[0112] Theorem 3 reduces the complexity of the non-convex matrix inequalities (20) and (21) through a series of model transformations. However, the matrices M1 and M2 still contain the non-linear term N T N, which makes the calculation difficult;

[0113] In this algorithm, we perform approximate linearization and solve it as a new unknown element; the effectiveness of the solution is judged by the performance constraint conditions in Theorem 2.

[0114] The beneficial effects of the present invention are as follows:

[0115] 1. The present invention discretizes the system, constructs an uncertain operator, and obtains an equivalent uncertain discrete-time model;

[0116] 2. The present invention solves the difficulty of applying the IQC theory to a system with an arbitrary initial state and gives an equivalent condition for exponential stability;

[0117] 3. The present invention combines the energy function of the uncertain operator and the IQC characteristics to optimize the H-infinity performance index of the measurement smoothness. Description of the Drawings

[0118] Figure 1 is the abstract drawing.

[0119] Figure 2 is the quarter-car model of the active suspension.

[0120] Figure 3 Schematic diagram of the feedback interconnection, where (a) is the model structure (7) and (b) is the model structure (9).

[0121] Figure 4 is the curve graph of the collision response changes of the body acceleration, suspension deflection, tire deflection, and actuator power.

[0122] Figure 5 is the curve graph of the collision response changes of the body acceleration, suspension deflection, tire deflection, and actuator power with time delay. Detailed Embodiments

[0123] The following describes the specific embodiments of the present invention with reference to the drawings so that those skilled in the art can better understand the present invention. It should be noted that in the following descriptions, when the detailed descriptions of known functions and designs may dilute the main content of the present invention, these descriptions will be omitted here.

[0124] Embodiment 1

[0125] Idea of a non-periodic sampling data H-infinity control method for an automotive active suspension system based on an uncertain discrete-time model.

[0126] First, consider an active vehicle suspension system with time-varying delay. Establish the dynamic equations of the sprung mass and the unsprung mass according to Newton's second law; establish a state-space model by defining state variables.

[0127] Second, to design the control law of the suspension system, consider four constraint conditions and reflect them in the input and output of the system. Establish an active vehicle suspension model with constraint conditions; make assumptions about the sampling data sequence, study the control law, and then obtain a discrete-time system model.

[0128] Third, based on Proposition 1 and Proposition 2, transform the discrete-time model into an equivalent feedback interconnection composed of an LTI system and a time-varying operator

[0129] According to the above idea, as Figure 1 shown, combined with a quarter-car active suspension model, a non-periodic sampling data H∞ control method for an automotive active suspension system based on an uncertain discrete-time model specifically includes the following steps:

[0130] Step 1: Establish an uncertain discrete-time model;

[0131] Step 2: Give the performance constraint conditions and stability conditions of the suspension system;

[0132] Step 3: Give the construction process of the equivalent feedback interconnection;

[0133] Step 4: Give an algorithm for solving the controller.

[0134] Example 2

[0135] The difference between this example and Example 1 is that the specific process of establishing the uncertain discrete-time model in Step 1 is as follows:

[0136] Consider an active vehicle suspension system with time-varying delay;

[0137] According to Newton's second law, the dynamic equations of the sprung mass and the unsprung mass are described as

[0138]

[0139] By defining the following state variables, such as:

[0140]

[0141] The disturbance input is The state \(x(t)=\text{col}\{x_1(t),x_2(t),x_3(t),x_4(t)\}\), and the dynamic equation (1) is rewritten as the state - space model shown below

[0142]

[0143] where,

[0144]

[0145] B w =[0 - 10c t / m u T .

[0146] Other steps and parameters are the same as those in Embodiment 1.

[0147] Embodiment 3

[0148] The difference between this embodiment and Embodiment 1 or 2 is that: the specific process of giving the performance constraint conditions and stability conditions of the suspension system in Step 2 is as follows:

[0149] Consider the following performance requirements:

[0150] ①Ride comfort; As is well - known, ride comfort is related to the body acceleration and is used as the performance output of the state - space model (2), denoted as \(z_1(t)\);

[0151] ②Suspension stroke; Considering the mechanical structure, the suspension stroke should not exceed the maximum allowable value \(y\) max , that is, \(\vert y\) s (t)-y u (t)\vert\leq y max ;

[0152] ③Road - holding; For driving safety, the contact between the tire and the road surface should be firm and continuous, and the dynamic load on the tire should be small, expressed as \(k\) t (y u (t)-y r (t))\leq9.8(m s +m u );

[0153] ④Actuator power limit; Due to the high power consumption of the active suspension system, the power applied to the actuator by the active suspension control should also be limited, that is, \(\vert u(t)\vert\leq u max ;

[0154] Considering the above four constraint conditions and reflecting them in the input - output of the system, the vehicle active suspension model is expressed as:

[0155] ​

[0156] where C1 = [-k s / m s 0 -c s / m s c s / m s , D1 = 1 / m s ,

[0157] Since there is a time-varying delay in the signal transmission process, the signal cannot be updated after sampling; to study the non-periodic sampling data control problem of the active suspension system, the following assumptions are made;

[0158] Assumption 1; Assume that {t k} is the sampling data sequence, satisfying where is the maximum sampling data interval;

[0159] Since the delay in the actual system is usually less than one sampling data interval, consider the case of τ k <h k ; According to the above description, the research control law is

[0160] for the design problem of state feedback control, where when k = 0, x t-1 = x init ;

[0161] On the sampling data interval, according to the characteristics of the input control law (4), solve the state equations at the sampling data time t k and the signal update time a k to obtain the discrete-time system model when x(t0) = x0:

[0162]

[0163] where

[0164] Other steps and parameters are the same as those in Embodiment 1 or 2.

[0165] Embodiment 4

[0166] The difference between this embodiment and one of Embodiments 1-3 is that the specific process of constructing the equivalent feedback interconnection given in Step 3 is as follows:

[0167] To analyze the discrete-time model (5), let h k = h0 + θ k , where is a constant, is the time-varying part; through the following two propositions, it is transformed into an equivalent feedback interconnection composed of an LTI system and a time-varying operator;

[0168] Proposition 1; Assume h k = h0 + θ k , where h0 is a fixed constant. By lifting the state and deforming the model, the discretized system is represented as the following uncertain discrete-time model:

[0169]

[0170] where,

[0171] By lifting the state Combined with the operator Prove that the discrete-time system model (5) and the uncertain discrete model (6) are equivalent;

[0172] Proposition 2; The uncertain discrete-time model (6) is converted into the following interconnection model

[0173]

[0174] where,

[0175] By defining new inputs and outputs, all uncertainty operators are represented by diagonal operators, and it can be seen that the uncertain discrete-time model (6) and the interconnection model (7) are equivalent;

[0176] Through the above two propositions, the discrete-time system model (5) is transformed into the interconnection model (7), paving the way for the application of the IQC theorem in subsequent work. On the other hand, note that the conversion between the interconnection model (7) and the discrete-time system model (5) is reversible. Therefore, the analysis of the discrete-time system model (5) is transformed into the analysis of the interconnection model (7).

[0177] Other steps and parameters are the same as those in one of Embodiments 1-3.

[0178] Embodiment 5

[0179] The difference between this embodiment and one of Embodiments 1-4 is that: the specific process of the algorithm for solving the controller given in Step 4 is as follows:

[0180] Analyze the stability and H-infinity performance of the system based on the IQC method;

[0181] When w(t) = 0, The interconnection model (7) is written as the following feedback interconnection:

[0182]

[0183] Since the feedback interconnection (8) cannot guarantee the condition of zero initial state, the IQC method cannot be directly used; to solve this problem, there is the following theorem;

[0184] Theorem 1; Assume that the matrix defined in the uncertain discrete-time model (6) satisfies Then the feedback interconnection (8) is exponentially stable if, for any Ψ(t k ) ∈ l1[0, ∞), the following interconnection is l2-stable. The interconnection with zero initial state:

[0185]

[0186] where, when k ≥ 1 and when k = 0 In the feedback interconnection (8) and are the same;

[0187] Δ1 and Δ2 are exponential functions of matrix A; to analyze the upper bound of the l2-norm of the uncertain operator, the following Schur decomposition of matrix A is performed:

[0188] U T AU = D + E, (10)

[0189] where U is a unitary matrix, D is a diagonal matrix, and E is a strictly upper triangular matrix; based on this decomposition, the following result is obtained;

[0190] Lemma 1; For any and κ ∈ [0, 1], the operator obtained in the feedback interconnection (8) satisfies the following IQC as shown below:

[0191]

[0192] where,

[0193] R1 = diag{R 11 , R 12 , R 13 , R 14}, Υ1 = diag{γ1I, γ1I, γ2I, γ1γ2I}, and α1 = λ max (A), α2 = λ max (-A);

[0194] Obtained with is correct; because for the operators Δ1 and Δ2, there exists Therefore, for any κ ∈ [0, 1],

[0195]

[0196] Therefore, the inequality (11) holds for any κ ∈ [0, 1];

[0197] Corollary 1. For any the operator given in Proposition 2 satisfies the following inequality constraints:

[0198]

[0199] where Υ2 = diag{γ1I, γ1I, γ2I, γ1γ2I, γ1I}, R2 = diag{R 21 , R 22 , R 23 , R 24 , R 25}, ε′ > 0, i = 1, 2, 3, 4, 5;

[0200] Lemma 2; Given a positive scalar γ, the system (7) satisfies if there exists R2 = diag{R 21 , R 22 , R 23 , R 24 , R 25}, such that

[0201]

[0202] where Θ1 = diag{P2, -P2}, Θ2 = diag{Υ2R2, I, -R2, -γ 2 I};

[0203] For the storage function V(x(t k )) = x(t k ) T P2x(t k ), multiply both sides of F(P2, R′1) by col{x(t k ), u(t k ), w(t k )}T and col{x(t k ), u(t k), w(t k )}, the inequality (13) is equivalent to

[0204] V(x(t k+1 )) - V(x(t k )) - γ 2 w T (t k )w(t k ) + z T (t k )z(t k ) + Υ2y T (t k )R2y(t k ) - u T (t k )R2u(t k ) < 0,

[0205] This means

[0206]

[0207] Since V(x(t k )) > 0 and x0 = 0, it is known that and V(x0) = 0, and further according to Corollary 1, it is known that holds;

[0208] Theorem 2; Assume Given a scalar γ > 0, and the state - feedback controller gain K exists in the form of the control law (4). If there exist scalar matrices Symmetric matrices and The following linear matrix inequalities are applicable:

[0209]

[0210] F(P2, R2) < 0, (16)

[0211]

[0212]

[0213] where S1 = [1 0], S2 = [0 1], N = diag(K, K), ρ = γ 2 w max + V(x(t0));

[0214] First, prove that the inequality (15) guarantees the exponential stability of the feedback interconnection (8); According to the KYP lemma, if the inequality (15) holds, then the following inequality

[0215]

[0216] Established, where is the transfer matrix of G in the feedback interconnection (9); thus there exists ε > 0 such that

[0217]

[0218] Established for all Lemma 1 shows that for any κ ∈ [0, 1], satisfies the IQC defined by Π; since is a linear causal, feedback interconnection is well - posed; thus, according to the IQC theorem, the transfer function (9) is l2 - stable, and then according to Theorem 1, the feedback interconnection (8) is exponentially stable;

[0219] Secondly, according to Lemma 2, when the inequality (16) is satisfied, there is

[0220] Thirdly, define and According to the proof of the input and output constraints, if the inequalities (17) and (18) hold, then |u(t k )| ≤ u max and |{z2(t k )} m | ≤ 1, m = 1, 2 hold.

[0221] Theorem 3; For the state - space model (2), assume Considering a scalar γ > 0, and a suitable ε > 0, if there exist a symmetric matrix M1 > 0, a scalar matrix and a matrix such that

[0222] W1 T Λ1W1 - εM1 < 0, (20)

[0223]

[0224]

[0225]

[0226] then the state - feedback controller gain matrix K that can satisfy the performance index can be obtained, K = [1 0]N[1 0] T , where M1 = [N0 -I] T [N 0 -I], M2 = [N 0 0 -I] T[N 0 0 -I], ρ = γ 2 w max + V(x(t0));

[0227] Combining the definitions in Propositions 1 and 2, let

[0228]

[0229] Substitute and be decomposed into where

[0230]

[0231] For a suitable ε > 0, we get

[0232]

[0233]

[0234] It is known that are respectively equal to Inequality (15) and Inequality (16); therefore, the controller finds N such that Inequalities (20)-(23) hold, and K = [1 0]N[1 0] T ;

[0235] To solve the controller gain matrix, the following iterative algorithm is given:

[0236]

[0237] Theorem 3 reduces the complexity of the non-convex matrix inequalities (20) and (21) through a series of model transformations; however, the matrices M1 and M2 still contain the non-linear term N T N, which makes the calculation difficult;

[0238] In this algorithm, we perform approximate linearization and solve it as a new unknown element; the effectiveness of the solution is judged by the performance constraint conditions in Theorem 2.

[0239] A non-periodic sampling data H∞ control method for an automotive active suspension system based on an uncertain discrete-time model of the present invention gives numerical simulation verification, indicating that when the automotive active suspension faces various stability influencing factors, the proposed control method can achieve faster stability and has better control performance, as follows:

[0240] In this part, we will illustrate the effectiveness of our design method based on the quarter-car active suspension model described in Section II; the model parameters, mu = 114 kg, m s = 973 kg, k t = 101115 N / m, k s = 42720 N / m, c t = 14.6 N s / m, c s = 1095 N s / m. In this paper, we assume z max = 0.08 m, u max = 1500 N, taking ρ = 1.

[0241] A. H-infinity performance analysis

[0242] Under the above conditions, when the sampling data interval and , through Algorithm 1, when ε = 10 -6 , the minimum value of the system H-infinity performance index is γ min = 7.26, which is less than γ min = 8.6758, and the controller gain matrix is K = 10 4 ×[-0.8249 1.1885 -0.8268 0.0359]. With the controller gain matrix K = 10 3 ×[0.7646 3.6362 -5.3292 0.0046], given the H-infinity performance index γ = 967.12, the upper limit interval of the sampling data can be obtained . In the same way, the upper limit interval of the delay For the sake of comparison, Table I gives the minimum performance H-infinity indexes under several upper bounds of sampling data. The γ guaranteed in this paper min is less than that under the same conditions. This shows that the method in this paper has less conservativeness.

[0243] Table 1

[0244] τ k = 0, γ under different sampling data intervals min

[0245]

[0246] B. Vehicle performance analysis

[0247] In this part, we will illustrate the usability of this method in solving vehicle performance problems. The performance study of the active suspension system mainly considers maneuverability, driving safety and comfort. This is reflected in the system as follows: the body acceleration |z1(t)| is as small as possible; the suspension deflection level |{z2(t)}1| = |x1(t) / y max| < 1; Tire deflection level |{z2(t)}2| = | k t x2(t) / (9.8(m s +m u )) | < 1, control input |u(t)| ≤ u max .

[0248] For the sake of comparison, the initial road surface displacement is considered in this paper.

[0249]

[0250] Where A = 60mm, L = 5m, V = 45km / h.

[0251] In the upper limit interval of the sampled data and τ k = 0ms, the passive case with K1 = 0 is considered; the continuous-time controller K2 = 10 4 ×[-8.992 -0.1447 -3.665 0.1491]; the sampled-data controller K3 = 10 3 ×[0.7646 3.6362 -5.3292 -0.0046], the sampled-data controller K4 = 10 4 ×[-0.8249 1.1885 -0.8268 0.0359]. The control effects of different controllers are as Figure 4 shown. Under the controller proposed in this paper, the body acceleration z1(t k ) is at a relatively low level; the suspension deflection level satisfies |{z2(t)}1| < 1 and is less than others; the tire deflection level satisfies |{z2(t)}2| < 1; the control input |u(t)| ≤ 1500N. In the continuous-time system, the system reaches stability faster than passive and sampled control.

[0252] C. Analysis of the influence of time delay on vehicle performance

[0253] This part studies the influence of delay on vehicle performance. To clearly explain the influence of delay on vehicle performance, according to Algorithm 1, assign For and the controllers are K = 10 3 ×[-8.5661 -0.9921 -1.4033 -0.7000] and K = 10 4 ×[-0.8423 -1.0920 -0.2756 -0.1134] respectively. The control effects of the controllers are as Figure 5 shown. The effects of both controllers are better than the dissipative type (u(t) ≡ 0). At the same time, the controller obtained without delay can achieve stability faster and can ensure better performance.

[0254] The other steps and parameters are the same as those in any one of Embodiments 1-4.

[0255] The non-periodic sampling data H∞ control problem of an automotive active suspension system based on an uncertain discrete model is studied. By lifting the states, an equivalent discrete-time model is obtained, and an uncertainty operator is constructed. For the uncertain model, a feedback interconnection model consisting of an LTI system and a time-varying operator is proposed. Under the guidance of the Bohl-Perron principle, the IQC theory is applied to the suspension system with non-zero initial states, and the conditions for the exponential stability of the system are given. Combining the IQC theory and the energy function, an optimized H-infinity performance index is given. The controller solving algorithm is given through model transformation, and the effectiveness of the method is verified by a quarter-vehicle model.

[0256] For any Ψ(t k ) ∈ l1[0, ∞), consider the inhomogeneous system corresponding to system (8) with zero initial state as follows:

[0257]

[0258] Let when k ≥ 1 and when k = 0. Substitute the above equations into the interconnected system (9) with zero initial state and compare it with system (A, 1), the solution of system (A, 1) can be obtained Therefore, the solutions of system (A, 1) and the interconnected system (9) with zero initial state are:

[0259]

[0260] where, when k ≥ 1 and when k = 0, Φ(t k ) = 0.

[0261] To prove the exponential stability of system (8) under the conditions of Theorem 1 by using the Bohl-Perron principle, we will prove that for any Ψ(t k ) ∈ l2[0, ∞) in system (A, 1), there exists

[0262] First, the property of ensures that the condition Ψ(t k ) ∈ l1[0, ∞) implies Therefore, when k ≥ 1, we have

[0263]

[0264] Combined with the definition of We can see that This means that

[0265] Secondly, if the interconnected system (9) with an initial state of zero is l2-stable, then there exists a constant c > 0 such that for any inequality holds. means that This means that From the expression of the interconnected system (9) with an initial state of zero, it can be seen that then is true. From the definition of the system (9), it can be seen that Since there exists an M that satisfies In the case of, from the definition of the generalized inverse matrix, it can be seen that there is a generalized inverse matrix such that Therefore Therefore, according to the relationship (A,2), we obtain

[0266]

[0267] This means that

[0268] Therefore, we have proved that for any Ψ(t k ) ∈ l2, the following is established According to the Bohl-Perron principle, the corresponding feedback interconnection (8) is exponentially stable.

[0269] The present invention may also have many other embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art can make various corresponding changes and deformations according to the present invention. However, these corresponding changes and deformations should all fall within the protection scope of the appended claims of the present invention.

Claims

1. A design method of an automotive suspension sampling controller based on a discrete system, characterized in that, It includes the following steps: Step 1: Establish an uncertain discrete-time model; Step 2: Give the performance constraints and stability conditions of the suspension system; Step 3: Give the construction process of the equivalent feedback interconnection; Step 4: Give the algorithm for solving the controller; The specific process of the said Step 1 is as follows: Consider an active vehicle suspension system with time-varying delay; According to Newton's second law, the dynamic equations of the sprung mass and the unsprung mass are described as: where m s is the sprung mass; m u is the unsprung mass; y s is the displacement of the sprung mass; y u is the displacement of the unsprung mass; y r is the road displacement input; c s is the damping coefficient of the spring damper; c t is the damping coefficient of the tire; k s is the suspension spring stiffness; k t is the tire stiffness; u is the force input by the actuator; By defining the following state variables, x1(t) = y s (t) - y u (t), x2(t) = y u (t) - y r (t), The disturbance input is The state x(t) = col{x1(t), x2(t), x3(t), x4(t)}, and the dynamic equation (1) is rewritten as the state - space model of the active suspension system shown below: Where, B u = [0 0 1 / m s -1 / m u T ,B w = [0 -10 c t / m u T ;​​ The specific process of the said Step 2 is as follows: Consider the following performance requirements: ①Ride comfort: Ride comfort is related to vehicle body acceleration and is used as the performance output of the state space model (2), denoted as z1(t); ② Suspension stroke: |y s (t) - y u (t)| ≤ y max ; ③ Road keeping: k t (y u (t) - y r (t)) ≤ 9.8(m s + m u ); ④ Actuator power limit: |u(t)| ≤ u max ; Considering the above four constraints and reflecting them in the input and output of the system, the vehicle active suspension model is expressed as: where C1 = [-k s / m s 0 - c s / m s c s / m s , D1 = 1 / m s , Due to the existence of time-varying delay in the signal transmission process, the signal cannot be updated after sampling; in order to study the non-periodic sampling data control problem of the active suspension system, the following assumptions are made; Hypothesis 1; Suppose {t k} is a sampled data sequence that satisfies where is the maximum sampling data interval; Since the delay in the actual system is usually less than one sampling data interval, consider τ k <h k ; According to the above description, the research control law is The design problem of state feedback control, where when k = 0, x t-1 = x init ; On the sampling data interval, solve for the sampling data time t according to the characteristics of the input control law (4) k and the signal update time a k for the state equation at, and obtain the discrete-time system model when x(t0) = x0: Among them, The specific process of the said Step 3 is as follows: To analyze the discrete-time system model (5), let h k = h0 + θ k , where is a constant and is the time-varying part; through the following two propositions, it is transformed into an equivalent feedback interconnection composed of an LTI system and a time-varying operator; Proposition 1; Assume h k = h0 + θ k , where h0 is a fixed constant. By raising the state and deforming the model, the discrete-time system model (5) is expressed as the following uncertain discrete-time model: Among them, By promoting the state Combined operator Prove that the discrete-time system model (5) and the uncertain discrete-time model (6) are equivalent; Proposition 2; The uncertain discrete-time model (6) is converted into the following interconnected model Among them, By defining new inputs and outputs, all uncertainty operators are represented by diagonal operators, and it can be seen that the uncertain discrete-time model (6) and the interconnection model (7) are equivalent; Through the above two propositions, the discrete-time system model (5) is transformed into the interconnection model (7), paving the way for the application of the IQC theorem in subsequent work; on the other hand, note that the conversion between the interconnection model (7) and the discrete-time system model (5) is reversible, so the analysis of the discrete-time system model (5) is transformed into the analysis of the interconnection model (7).

2. A design method of an automotive suspension sampling controller based on a discrete system according to claim 1, characterized in that: The specific process of giving the algorithm for solving the controller in the said Step 4 is as follows: Analyze the stability and H-infinity performance of the system based on the IQC method; When w(t) = 0, The interconnection model (7) is written as the following feedback interconnection: Since the feedback interconnection (8) cannot guarantee the condition that the initial state is zero, the IQC method cannot be directly used; to solve this problem, there is the following theorem; Theorem 1; Assume that for the matrices in the uncertain discrete-time model (6) satisfy Then the feedback interconnection (8) is exponentially stable if, for any Ψ(t k ) ∈ l1[0, ∞), the following interconnection is l2-stable; the interconnection with zero initial state: wherein, when k≥1 and when k = 0 in the feedback interconnection (8) and are the same; △1 and △2 are exponential functions of matrix A; in order to analyze the upper bound of the l2 norm of the uncertainty operator, the following Schur decomposition is performed on matrix A: U T AU = D + E, (10) Where U is a unitary matrix, D is a diagonal matrix, and E is a strictly upper triangular matrix; based on this decomposition, the following results are obtained; Lemma 1: For any and κ ∈ [0, 1], the operator obtained in the feedback interconnection (8) satisfies the following IQC as shown below: Where, R1 = diag{R 11 , R 12 , R 13 , R 14}, γ1 = diag{γ1I, γ1I, γ2I, γ1γ2I}, and α1 = λ max (A), α2 = λ max (-A); Obtain and is correct; because for operators △1 and △2, there exists Therefore, for any κ ∈ [0, 1], Therefore, the inequality (11) holds for any κ ∈ [0, 1]; Corollary 1: For any the operator given in Proposition 2 satisfies the following inequality constraints: where, γ2 = diag{γ1I, γ1I, γ2I, γ1γ2I, γ1I}, R2 = diag{R 21 , R 22 , R 23 , R 24 , R 25}, ε′ > 0, i = 1, 2, 3, 4, 5; Lemma 2; Given a positive scalar γ, the interconnection model (7) satisfies If there exists R2 = diag{R 21 , R 22 , R 23 , R 24 , R 25}, Make Among them, Θ1 = diag{P2, -P2}, Θ2 = diag{Υ2R2, I, -R2, -γ 2 I}; For the storage function V(x(t k )) = x(t k ), T P2x(t k ), multiply both sides of F(P2, R1′) by col{x(t k ), u(t k ), w(t k )}T and col{x(t k ), u(t k ), w(t k ), and the resulting inequality (13) is equivalent to V(x(t k+1 )) - V(x(t k )) - γ 2 w T (t k )w(t k ) + z T (t k )z(t k ) +Υ2y T (t k )R2y(t k )-u T (t k )R2u(t k )<0, This means Since V(x(t k )) > 0 and x0 = 0, it is known that and V(x0) = 0. Furthermore, according to Corollary 1, it is known that holds; Theorem 2; Suppose Given a scalar γ > 0, and a state - feedback controller gain K in the form of control law (4), if there exist scalar matrices symmetric matrices and the following linear matrix inequalities are applicable: F(P2, R2) < 0, (16) where \(S1 = [1\ 0]\), \(S2 = [0\ 1]\), \(N=\text{diag}(K, K)\), \(\rho=\gamma\) 2 w max +V(x(t0)); First, prove that the inequality (15) guarantees the exponential stability of the feedback interconnection (8); according to the KYP lemma, it is found that if the inequality (15) holds, then there is the following inequality Established, where is the transfer matrix of G in the feedback interconnection (9); thus there exists ε > 0 such that Based on all Lemma 1 shows that for any κ ∈ [0, 1], satisfies the IQC defined by ∏; Since is a linear causal, feedback interconnection is well-posed; Therefore, according to the IQC theorem, the transfer matrix (9) is l2-stable, and then according to Theorem 1, the feedback interconnection (8) is exponentially stable; Secondly, according to Lemma 2, when the inequality (16) is satisfied, we have Third, define and According to the proof based on the input and output constraints, it is known that if inequalities (17) and (18) hold, then |u(t k )| ≤ u max and |{z2(t k )} m | ≤ 1, where m = 1, 2 holds; Theorem 3; State-space model (2), assumption Given a scalar γ > 0, and a suitable ε > 0, if there exist symmetric matrices M1 > 0, scalar matrices and matrices such that Then the state feedback controller gain matrix K that satisfies the performance index constraint conditions is obtained, K = [1 0]N[1 0] T , where M1 = [N 0 -I] T [N 0 -I], M2 = [N 0 0 -I] T [N 0 0 -I], ρ = γ 2 w max + V(x(t0)); Combining the definitions in Propositions 1 and 2, let will and be decomposed into wherein For a suitable ε > 0, obtain From Lemma 1, we know that are respectively equal to Inequality (15) and Inequality (16); thus, the controller finds N such that Inequalities (20)-(23) hold, and K = [1 0]N[1 0] T ; To solve the controller gain matrix, the following iterative algorithm is given: S1: Give the upper limit interval of the sampling data and the upper limit interval of the delay an initial value γ > 0 and a scalar ε = 1; S2: Set N T Let N = M, solve the LMIs (20)-(23) to obtain the controller gain matrix, where K = [1 0]N[1 0] T ; S3: If there is a feasible solution for the LMIs (15)-(18) and K obtained above; S4: γ = γ - 0.01, repeat Steps 2-3; S5: else; S6: Let ε = ε * 0.1, repeat 2-5; S7: endif; S8: Output the controller gain matrix K and the H∞ performance index γ; Theorem 3 reduces the complexity of the non-convex matrix inequalities (20) and (21) through a series of model transformations; however, the matrices M1 and M2 still contain the non-linear term N T N, which makes the calculation difficult; In this algorithm, approximate linearization is performed and it is solved as a new unknown element; the effectiveness of the solution is judged by the performance constraint conditions in Theorem 2.

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