Memory dynamic output feedback control method for input time-delay automotive suspension

Abstract

The invention relates to the technical field of active suspension system control, in particular to a memory dynamic output feedback control method for an input time-delay automotive suspension. Comprising the steps that a kinetic equation of the automobile active suspension system is established, and an equivalent kinetic equation is given under a memory operator; giving a linear time-invariant system form induced by integral quadratic constraint; giving a memory dynamic output feedback controller structure under an integral quadratic constraint method; through model transformation, a closed-loop system form under feedback of the controller is given; under an integral quadratic constraint framework, stability conditions of the input time-delay automobile active suspension system under output constraint and input constraint are given; providing a design method of the memory dynamic output feedback controller of the automobile active suspension system; and solving a memory dynamic output feedback controller gain matrix of the automobile active suspension system. According to the method, the model processing process is greatly simplified, the calculation complexity is reduced, and stability control over the input time-delay automobile active suspension is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of active suspension system control, and specifically belongs to a memory dynamic output feedback control method for an automotive suspension with input time delay. Background Art

[0002] In the field of automotive industrial design, a suspension system consists of shock absorbers, springs, and links. The links are one of the key components connecting the vehicle frame and the axle. It can effectively isolate, absorb, and dissipate the transient vibrations caused by the occupants and the road, thus providing a more comfortable riding experience. With the help of a braking device, the active suspension system of a vehicle can actively modify its elasticity, damping, and output driving power in response to feedback signals. This greatly improves the ability of the suspension to adapt to uneven road surfaces. For suspension systems, many control strategies have been proposed, including H ∞ robust control, fuzzy control, sliding mode control, sampling control, adaptive control, etc. These strategies have further improved the vehicle's maneuverability, driving smoothness, and safety, and have received increasing attention from academia and industry.

[0003] In a vehicle active suspension system, the use of hydraulic or pneumatic braking mechanisms for guidance and the time-varying input delay caused by communication delays can reduce control performance, have a negative impact on the system performance in practical applications, and in some cases, even more seriously lead to instability problems. However, it is difficult for traditional controllers to balance delay compensation and design complexity. The method proposed by Hongyi Li (Journal: IEEE Transactions on Industrial Electronics; Authors: Hongyi Li, Xingjian Jing, Hamid Reza Karimi; Publication Time: 2024; Article Title: Output-Feedback-Based H∞ Control for Vehicle Suspension Systems With Control Delay; Pages: 436-446) presents a dynamic output feedback controller design scheme to ensure various vehicle active suspension performance limitations while achieving asymptotic stability of the suspension system. The method proposed by Hyun Duck Choi (Journal: IEEE Transactions on Fuzzy Systems; Authors: Hyun Duck Choi, Choon Ki Ahn, Peng Shi, Ligang Wu, Myo Taeg Lim; Publication Time: 2017; Article Title: Dynamic Output-Feedback Dissipative Control for T–S Fuzzy Systems With Time-Varying Input Delay and Output Constraints; Pages: 511-526) is based on (Q, S, R)-α-dissipativity. For a Takagi–Sugeno (T–S) fuzzy system with time-varying input delay and output constraints, the proposed controller is called a (Q, S, R)-α-dissipative output feedback fuzzy controller. It takes into account abstract energy, storage functions, and perturbation attenuation supply rates and provides a unified framework that can incorporate H ∞The existing results of passivity-based controllers are considered as special cases of T–S fuzzy systems with time-varying input delays and output constraints. The method of Jing Zhao (Journal: IEEE / ASME Transactions on Mechatronics; Authors: Jing Zhao, Pak Kin Wong, Wenfeng Li, Meisam Ahmadi Ghadikolaeia, Zhengchao Xie; Publication Time: 2022; Article Title: Reliable Fuzzy Sampled-Data Control for Nonlinear Suspension Systems Against Actuator Faults; Pages: 5518-5528) uses a Lyapunov function candidate to obtain sufficient conditions that simultaneously satisfy the system stability and performance requirements, and designs a sampled-state feedback and a static output feedback controller. All of the above cases lack an appropriate application of the real-time relationship between the controller output and the delayed input reaching the actuator. Many studies have demonstrated the effectiveness of the integral quadratic constraint technique in active suspension systems, especially in the field of vehicle control.

[0004] The dynamic integral quadratic constraint method is also a powerful tool for solving stability problems. In addition, designing or selecting appropriate dynamic integral quadratic constraint multipliers can greatly reduce the conservatism of controller design and describe uncertainties more accurately than static integral quadratic constraint multipliers. Summary of the Invention

[0005] To solve the stability control technology problem of an automotive suspension with input time delay, the present invention proposes a memory dynamic output feedback control method for an automotive suspension with input time delay, thereby greatly simplifying the model processing process, reducing the computational complexity, and achieving the stability control of the input time delay of an automotive active suspension.

[0006] A memory dynamic output feedback control method for an automotive suspension with input time delay provided by the present invention includes the following steps.

[0007] Step 1: Establish the dynamic equation of an automotive active suspension system and give an equivalent dynamic equation under the memory operator.

[0008] Step 2: Give the form of a linear time-invariant system induced by integral quadratic constraints.

[0009] Step 3: Considering the existence of time delay, i.e., input time delay, in the control, give the structure of a memory dynamic output feedback controller induced by integral quadratic constraints.

[0010] Step 4, through model transformation, present the form of the closed-loop system composed of the input-delay automotive active suspension system, the linear time-invariant system induced by integral quadratic constraints, and the memory dynamic output feedback controller;

[0011] Step 5, under the induction of integral quadratic constraints, give the stability conditions of the input-delay automotive active suspension system under output constraints and input constraints;

[0012] Step 6, give the design method of the memory dynamic output feedback controller for the input-delay automotive active suspension system;

[0013] Step 7, solve the gain matrix of the memory dynamic output feedback controller for the input-delay automotive active suspension system.

[0014] Furthermore, in Step 1, the sprung mass and unsprung mass of a quarter-car are m s and m u , the gravitational constant is g, the stiffness and damping of the automotive active suspension system are c s and k s , the compressibility and damping of the pneumatic tire are c t and k t , the displacements of the sprung mass and unsprung mass are z s (t) and z u (t), the road surface displacement input is z r (t), and the braking force of the actuator is u(t);

[0015] According to Newton's second law, under the suspension motion range and tire dynamic load limit, the dynamic equation of the quarter-car active suspension system is

[0016]

[0017] where represents the input function with time-varying delay d(t) at time t, and r represent the maximum delay and its maximum derivative respectively, and for any t ≥ 0, satisfy and

[0018] The output for minimizing the body acceleration is set as Due to the limitation of the mechanical structure, the stroke of the automotive active suspension cannot exceed the upper limit of the suspension dynamic deflection z max , that is, |z s (t) - z u (t)| ≤ z max ; Since the contact between the tire and the road surface is firm and continuous, the tire load relationship can be expressed as (z u (t) - z r (t))kt <(m s +m u )g, let another controlled output be Due to the limited power of the actuator, the active control force of the vehicle active suspension should be limited within u max , that is, |u(t)| ≤ u max ;

[0019] Let the four state components be x1(t) = z s (t) - z u (t), x2(t) = z u (t) - z r (t), Let the state variable be x(t) = [x1(t), x2(t), x3(t), x4(t)] Τ , and the measurable output be y(t) = [x1(t), x3(t), x4(t)] Τ , then the dynamic equation (1) of the quarter - car active suspension system is transformed into the following system

[0020]

[0021] where x(0) = x0 is the initial state, and

[0022]

[0023] Define and Then the system (2) can be expressed as the following system

[0024]

[0025] where (A, B, C) is controllable and observable, and

[0026] Furthermore, in step 2, assume that is a proper rational function, called the "multiplier", satisfying Π = ψ~Wψ, where and If the inequality

[0027]

[0028] holds for T ≥ 0, then the two signals and satisfy the integral quadratic constraint IQC defined by the multiplier Π, and (ψ, W) is a hard IQC factorization of Π, where denotes the filtered output driven by an input \((v,\eta)\) with zero initial conditions; furthermore, a bounded causal operator \(\varphi:\) satisfies the condition given by where the frequency-domain multiplier can be factored as and For any each multiplier satisfies \(\Pi\) 11,k (j\omega)>0 and \(\Pi\) 22,k (j\omega)<0; furthermore, for any \(k\in\{1,2,\ldots,N\}\) λ}, \(\Pi\) k has a spectral factorization of the form for a spectral factorization;

[0029] Therefore, the system can be represented as a linear time-invariant system

[0030]

[0031] where denotes the state vector of the operator with \(x\) ψ (0)=0, and and denotes the output vector of the operator , and \(n\) α =2n u ; furthermore, for any \(k\in\{1,2,\ldots,N\}\) λ}, the output matrix of the IQC-induced system (5) is

[0032]

[0033] where and have appropriate dimensions.

[0034] Furthermore, in step 3, corresponding to the application of the IQC method, a memory dynamic output feedback controller with input delay is proposed for the dynamic equation (1) as follows

[0035]

[0036] where is the state vector of the memory dynamic output feedback controller, and \(n\) c is to be determined; since the input signal \(u(t)\) and can be directly measured in real time, then the state vector \(x\) ψ (t) of the IQC-induced system (5) can also be computed.

[0037] Furthermore, in step 4, by combining the system (3), the IQC induction system (5), and the memory dynamic output feedback controller (6), for any k ∈ {1, 2,..., N λ}, the closed-loop system is as follows

[0038]

[0039] where and the relevant system matrices are

[0040]

[0041] Furthermore, in step 5, considering the closed-loop system (7), given scalars and q = 1, 2, if there exist positive definite matrices and scalars such that

[0042]

[0043] where

[0044]

[0045] then the closed-loop system (7) with delay d(t) is stable, where and and minimizing the H q performance index under the output constraint |{z2(t)} 2,max | ≤ {z q , q = 1, 2, t ≥ 0, z 2,max = [1 1] Τ , the maximum actuator control force constraint in |u(t)| ≤ u ∞ is under zero initial conditions, and the disturbance energy is in the range of w max = (ρ - V(0)) / γ max , where 2

[0046] Furthermore, in step 6, considering the closed-loop system (7), given scalars and q = 1, 2, if there exist positive definite matrices matrix and scalars such that

[0047]

[0048] ​

[0049] Among them

[0050]

[0051] Then, there exists a dynamic output feedback controller such that the closed-loop system (7) is stable; and the ideal dynamic output feedback controller control gain in the form of the memory dynamic output feedback controller (6) is given by the following:

[0052]

[0053] And the invertible matrices M and N satisfy

[0054]

[0055] Furthermore, in step 7, the input parameters r, ρ; the calculation method is as follows

[0056] a. Solve the following minimization problem

[0057]

[0058] b. Mark the optimal solution

[0059] c. Through Solve the minimization problem (20);

[0060] d. Solve the optimal solution Let γ l = γ, l = l + 1;

[0061] e. When γ l -γ l-1 < -0.1 and t min < 0, repeat steps c - e; otherwise execute step f;

[0062] f. End the loop and output the matrix

[0063] g. Return the controller gain matrices A c , B c0 , B c1 , B c2 , C c .

[0064] The present invention has the following beneficial effects

[0065] 1. Under the induction of dynamic integral quadratic constraint, the proposed dynamic output feedback controller design method of the present invention can characterize the uncertainty between the controller output and the delayed input reaching the actuator. To control the active suspension system, data from the delay loop can be incorporated into the memory dynamic output feedback controller;

[0066] 2. The present invention takes into account multiple performance constraints of the actual active suspension system, so as to improve the suspension performance while meeting the actual situation when the memory dynamic output feedback controller controls the active suspension system. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 is a flowchart of the method of the present invention;

[0068] Figure 2 is a schematic diagram of a quarter-car active suspension system involved in the present invention;

[0069] Figure 3 is the suspension response of the controller gain matrix obtained by the method of the present invention and the methods of Hongyi Li and Hyun Duck Choi;

[0070] Figure 4 is the suspension performance of the controller gain matrix obtained by the method of the present invention and the methods of Hongyi Li and Hyun Duck Choi;

[0071] Figure 5 is the actuator force of the controller gain matrix obtained by the method of the present invention and the methods of Hongyi Li and Hyun Duck Choi;

[0072] Figure 6 is the suspension response of the controller gain matrix obtained by the method of the present invention and the method of Jing Zhao;

[0073] Figure 7 is the suspension performance of the controller gain matrix obtained by the method of the present invention and the method of Jing Zhao;

[0074] Figure 8 is the actuator force of the controller gain matrix obtained by the method of the present invention and the method of Jing Zhao. DETAILED DESCRIPTION OF THE INVENTION

[0075] The following further clarifies the present invention in conjunction with the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and not to limit the scope of the present invention.

[0076] The present invention provides a memory dynamic output feedback control method for an input time-delay automotive suspension, as Figure 1 shown, mainly including:

[0077] Step 1: Establish the dynamic equation of the automotive active suspension system and give the equivalent dynamic equation under the memory operator;

[0078] Step 2: Give the form of the linear time-invariant system induced by the integral quadratic constraint;

[0079] Step 3: Considering the time delay existing in the control, i.e., the input time delay, give the structure of the memory dynamic output feedback controller induced by the integral quadratic constraint;

[0080] Step 4: Through model transformation, give the form of the closed-loop system composed of the input time-delay automotive active suspension system, the linear time-invariant system induced by the integral quadratic constraint, and the memory dynamic output feedback controller;

[0081] Step 5: Under the induction of the integral quadratic constraint, give the stability conditions of the input time-delay automotive active suspension system under output constraints and input constraints;

[0082] Step 6: Give the design method of the memory dynamic output feedback controller for the input time-delay automotive active suspension system;

[0083] Step 7: Solve the gain matrix of the memory dynamic output feedback controller for the input time-delay automotive active suspension system.

[0084] The following elaborates on the specific implementation process of the present invention in detail.

[0085] Step 1: As Figure 2 shown, the sprung mass and unsprung mass of a quarter-car are m s and m u , the gravitational constant is g, the stiffness and damping of the automotive active suspension system are c s and k s , the compressibility and damping of the pneumatic tire are c t and k t , the displacements of the sprung mass and unsprung mass are z s (t) and z u (t), the road surface displacement input is z r (t), and the braking force of the actuator is u(t);

[0086] According to Newton's second law, under the suspension motion range and tire dynamic load limit, the dynamic equation of the quarter-car active suspension system is

[0087]

[0088] where represents the input function with a time-varying delay d(t) at time t, The maximum delay and its maximum derivative are denoted by \(r\) and \(r\) respectively, and for any \(t\geq0\), they satisfy and

[0089] The output that minimizes the vehicle body acceleration is set as Due to the limitations of the mechanical structure, the stroke of the vehicle's active suspension cannot exceed the upper limit \(z\) of the suspension dynamic deflection max , that is, \(\vert z\) s (t)-z u (t)\vert\leq z max ; Since the contact between the tire and the road surface is firm and continuous, the tire load relationship can be expressed as \((z\) u (t)-z r (t))k t <\((m\) s +m u )g. Let another controlled output be Due to the limited power of the actuator, the active control force of the vehicle's active suspension should be limited within \(u\) max , that is, \(\vert u(t)\vert\leq u max ;

[0090] Let the four state components be \(x_1(t)=z\) s (t)-z u (t), \(x_2(t)=z\) u (t)-z r (t), Let the state variable be \(x(t)=[x_1(t),x_2(t),x_3(t),x_4(t)] Τ , and the measurable output be \(y(t)=[x_1(t),x_3(t),x_4(t)] Τ , then the dynamic equation (1) of the quarter-vehicle active suspension system is transformed into the following system

[0091]

[0092] where \(x(0)=x_0\) is the initial state, and

[0093]

[0094] Define and Then the system (2) can be expressed as the following system

[0095]

[0096] where \((A,B,C)\) is controllable and observable, and

[0097] Step 2: Assume is a proper rational function, called the "multiplier", satisfying Π = ψ ∼ Wψ, where and If the inequality for T ≥ 0

[0098]

[0099] holds, then the two signals and satisfy the integral quadratic constraint IQCs defined by the multiplier Π, and (ψ, W) is a hard IQC factorization of Π, where denotes the filtered output driven by the input (v, η) with zero initial conditions; furthermore, a bounded causal operator φ : satisfies the one defined by where the frequency-domain multiplier can be factored as and For any each multiplier satisfies Π 11,k (jω) > 0 and Π 22,k (jω) < 0; furthermore, for any k ∈ {1, 2,..., N λ}}, Π k all have a of the form spectral factorization;

[0100] Therefore, the system can be represented as a linear time-invariant system

[0101]

[0102] where denotes the state vector of the operator ψ with x (0) = 0, and denotes the output vector of the operator and n α = 2n u ; furthermore, for any k ∈ {1, 2,..., N λ}}, the output matrix of the IQC-induced system (5) is

[0103]

[0104] where and have appropriate dimensions.

[0105] Step 3: Corresponding to the application of the IQC method, a memory dynamic output feedback controller structure with input delay is proposed for the dynamic equation (1) as follows

[0106]

[0107] where is the state vector of the memory dynamic output feedback controller, and n c is to be determined; since the input signal u(t) and can be directly measured in real time, then the state vector x ψ (t) of the IQC-induced system (5) can also be calculated.

[0108] Step 4: By combining the system (3), the IQC-induced system (5) and the memory dynamic output feedback controller (6), for any k ∈ {1, 2,..., N λ} the closed-loop system is as follows

[0109]

[0110] where and the relevant system matrices are

[0111]

[0112] Step 5: Considering the closed-loop system (7), given scalars and q = 1, 2, if there exist positive definite matrices and scalars such that

[0113]

[0114] where

[0115]

[0116] then the closed-loop system (7) with delay d(t) is stable, where and and under the output constraint |{z2(t)} q | ≤ {z 2,max} q , q = 1, 2, t ≥ 0, z 2,max = [1 1] Τ minimize the H ∞ performance index, the maximum actuator control force constraint in |u(t)| ≤ u max under zero initial conditions, the disturbance energy is in w max = (ρ - V(0)) / γ 2 range, where

[0117] Step 6: Considering the closed-loop system (7), given scalars and \(q = 1,2\), if there exists a positive definite matrix matrix and a scalar such that

[0118]

[0119] where

[0120] Then, there exists a dynamic output feedback controller such that the closed-loop system (7) is stable; and an ideal dynamic output feedback controller of the form of the memory dynamic output feedback controller (6) has its control gains given by the following:

[0121]

[0122] and invertible matrices \(M\) and \(N\) satisfy

[0123]

[0124] Step 7: Input parameters \(r\), \(\rho\); The calculation method is as follows,

[0125] a. Solve the following minimization problem

[0126]

[0127] b. Mark the optimal solution

[0128] c. Solve the minimization problem (20) through ;

[0129] d. Solve the optimal solution Let \(\gamma\) l \(=\gamma\), \(l = l + 1\);

[0130] e. When \(\gamma\) l \(-\gamma\) l-1 \(< -0.1\) and \(t\) min \(< 0\), repeat steps c - e; otherwise execute step f;

[0131] f. End the loop and output the matrix

[0132] g. Return the controller gain matrices \(A\) c , \(B\) c0 , \(B\) c1 , \(B\) c2 , \(C\) c .

[0133] This calculation method addresses the impact of continuous-time input delay on the automotive active suspension system, improves the performance of the active suspension system within the framework of dynamic integral quadratic constraints, and proposes an iterative correlation-based controller solution algorithm.

[0134] To verify the effectiveness of the proposed theoretical results, the specific numerical values of the relevant parameters of a quarter-car active suspension used in the simulation experiment are as follows:

[0135] According to the quarter-car active suspension system (2) of the present invention, the methods of Hongyi Li, Jing Zhao, and Hyun Duck Choi, the relevant system parameters can be obtained as shown in Table 1.

[0136] Table 1 Active suspension system parameters

[0137]

[0138] Let r = 0.9. For the first set of values in Table 1, the controller gain matrix in the method of Hyun Duck Choi can be obtained as K Choi = [91.2579 -50.7782 -40.6294 3.7329], and the controller gain matrix in the method of Hongyi Li is

[0139]

[0140] Next, through the analysis of the memory dynamic output feedback control results of the application embodiments of the present invention, the present invention will be further described and explained.

[0141] Let r = 0.9, ρ = 1. For the parameter settings of the integral quadratic constraint, we select two integral quadratic constraint multipliers That is

[0142]

[0143] Where And c1 is an arbitrary positive real number such that c1 < 2k1, Where ε and δ are both arbitrary small positive real numbers. To verify the effectiveness of the method of the present invention in the suspension system, let c1 = 1.1, ε = 10 -8 , δ = 10 -4 . It can be obtained that

[0144]

[0145]

[0146] Therefore, the system matrix of the system (5) induced by the integral quadratic constraint can be expressed as

[0147]

[0148] Furthermore, the memory dynamic output feedback controller gain matrix is B c0 , B c1 , B c2 , C c , that is

[0149]

[0150]

[0151] As described in Table 2, the decision variables involved in the memory dynamic output feedback controller design method proposed in this paper, the methods of Hyun Duck Choi and Hongyi Li are 212 < 462 < 593 respectively. When it comes to solving the controller corresponding to the first set of parameter values in Table 1, the running time of the memory dynamic output feedback controller design method proposed in this paper, 2.58 s, is less than the running time of the method of Hongyi Li, 39.44 s. Compared with the Lyapunov method, the computational complexity of the memory dynamic output feedback controller design method using two integral quadratic constraint multipliers is much lower.

[0152] Table 2: Number of decision variables required for solving the controller by different control methods

[0153]

[0154] To verify the effect of the memory dynamic output feedback controller proposed in the present invention in solving the delay in the system (2), the pulse input road surface disturbance signal is selected as

[0155]

[0156] At Figure 3 , Figure 4 and Figure 5 show the suspension response control effects related to the memory dynamic output feedback controller proposed in this paper, the controller designed by the method of Hyun Duck Choi and the controller designed by the method of Hongyi Li. At Figure 3Among them, the light blue solid line, dark blue dashed line, and red solid line represent the positions of the ground, tire, and vehicle body, respectively. Obviously, the memory dynamic output feedback controller proposed by the present invention can enable the vehicle body of the active suspension to converge to road surface vibrations quickly and smoothly, and the change of the control force is relatively smoother. Obviously, the designed memory dynamic output feedback controller is superior to the methods of Hyun Duck Choi and Hongyi Li.

[0157] For further comparison, consider the state feedback controller gain matrix in the method of Jing Zhao and the output feedback controller gain matrix As for the relevant responses of the suspension Figure 6 shown, the vehicle body acceleration and tire deflection are respectively as Figure 7 and Figure 8 shown. Obviously, the control effect of the memory dynamic output feedback controller proposed by the present invention is far superior to the state feedback and output feedback control schemes in the method of Jing Zhao.

[0158] The simulation results show that the controller can effectively handle the time-varying delay of the networked active suspension system while maintaining the best ride comfort. The comparative study confirms the practicability of this method in the automotive suspension system and further verifies its practicability in adapting to road surface disturbances and time delays.

[0159]

[0160] To evaluate the applicability of the controller, select the second set of parameter values in Table 1, and set r = 0.9, ρ = 1. Table 2 shows the comparison of the H ∞ performance index γ min for the active suspension system (2) of the memory dynamic output feedback controller proposed by the present invention, the methods of Hongyi Li and Hyun Duck Choi. Table 3 gives the comparison of the number of decision variables required to solve the controller in the memory dynamic output feedback controller proposed by the present invention, the methods of Hongyi Li and Hyun Duck Choi. This can explain that the proposed memory dynamic output feedback controller has lower computational complexity and faster computational speed when solving the controller gain matrix.

[0161] Compared with the prior art, the advantages of the present invention are as follows: Under the dynamic integral quadratic constraint framework, the proposed dynamic output feedback controller design method can characterize the uncertainty between the controller output and the delayed input reaching the actuator. For controlling an active suspension system, data from the delay loop can be incorporated into the memory dynamic output feedback controller, greatly simplifying the model processing procedure and reducing the computational complexity; multiple performance constraints of the actual active suspension system are taken into account so as to improve the suspension performance while meeting the actual situation when the memory dynamic output feedback controller controls the active suspension system; a specific algorithm for solving the controller gain matrix is given.

Claims

1. A memory dynamic output feedback control method for an input time-delay automotive suspension, characterized in that, including the following steps Step 1: Establish the dynamic equation of the automotive active suspension system and give the equivalent dynamic equation under the memory operator; Step 2: Give the form of the linear time-invariant system induced by the integral quadratic constraint; Step 3: Considering the time delay existing in the control, i.e., the input time delay, give the structure of the memory dynamic output feedback controller induced by the integral quadratic constraint; Step 4: Through model transformation, give the form of the closed-loop system composed of the input time-delay automotive active suspension system, the linear time-invariant system induced by the integral quadratic constraint, and the memory dynamic output feedback controller; Step 5: Under the induction of the integral quadratic constraint, give the stability conditions of the input time-delay automotive active suspension system under output constraints and input constraints; Step 6: Give the design method of the memory dynamic output feedback controller for the input time-delay automotive active suspension system; Step 7: Solve the gain matrix of the memory dynamic output feedback controller for the input time-delay automotive active suspension system.

2. The memory dynamic output feedback control method for an input time-delay automotive suspension according to claim 1, wherein In Step 1, The sprung mass and unsprung mass of a quarter-car are \(m\) s and \(m\) u , the gravitational constant is \(g\), the stiffness and damping of the vehicle's active suspension system are \(c\) s and \(k\) s , the compressibility and damping of the pneumatic tire are \(c\) t and \(k\) t , the displacements of the sprung mass and unsprung mass are \(z\) s (t) and \(z\) u (t), the road surface displacement input is \(z\) r (t), and the braking force of the actuator is \(u(t)\); According to Newton's second law, under the limitations of the suspension motion range and tire dynamic load, the dynamic equation of the quarter-car active suspension system is wherein represents an input function with a time-varying delay d(t) at time t, and r respectively represent the maximum delay and its maximum derivative, satisfying, for any t ≥ 0, and The output for minimizing the vehicle body acceleration is set to Due to the limitations of the mechanical structure, the stroke of the vehicle active suspension cannot exceed the upper limit z of the suspension dynamic deflection max , that is, |z s (t) - z u (t)| ≤ z max ; Since the contact between the tire and the road surface is firm and continuous, the tire load relationship is expressed as (z u (t) - z r (t))k t < (m s + m u ). Let another controlled output be Due to the limited power of the actuator, the active control force of the vehicle active suspension is limited within u max , that is, |u(t)| ≤ u max ; Let the four state components be \(x_1(t)=z s (t)-z u (t), \(x_2(t)=z u (t)-z r (t), Let the state variable be \(x(t)=[x_1(t),x_2(t),x_3(t),x_4(t)] Τ , then the output is \(y(t)=[x_1(t),x_3(t),x_4(t)] Τ , then the dynamic equation (1) of the quarter-car active suspension system is transformed into the following system where \(x(0)=x_0\) is the initial state, and Definition and Then the system (2) is expressed as the following system where (A, B, C) is controllable and observable, and 3. A memory dynamic output feedback control method for an input time-delay automotive suspension according to claim 2, characterized in that In Step 2, Hypothesis is a rational function, called the "multiplier", satisfying Π = ψ ∼ Wψ, where and If the inequality for T ≥ 0 hold, then the two signals and satisfy the integral quadratic constraint (IQC) defined by the multiplier Π, and (ψ, W) is a hard IQC factorization of Π, where denotes the filtered output driven by the input (v, η) with zero initial conditions; In addition, a bounded causal operator satisfies the condition that where the frequency-domain multiplier can be decomposed into and For any each multiplier satisfies Π 11,k (jω) > 0 and Π 22,k (jω) < 0; in addition, for any k ∈ {1, 2,..., N λ} k has a spectral decomposition in the form of and a spectral decomposition; Therefore, the system can be represented as a linear time-invariant system where denotes the state vector with x ψ (0) = 0, and and denotes the output vector of the operator , and n α = 2n u ; in addition, for any k ∈ {1, 2,..., N λ} the output matrix of the IQC-induced system (5) is Among them and have appropriate dimensions.

4. The memory dynamic output feedback control method for an input time-delay automotive suspension according to claim 3, characterized in that, In Step 3, Corresponding to the application of the IQC method, a memory dynamic output feedback controller with input delay is proposed for the dynamic equation (1) as follows where is the state vector of the memory dynamic output feedback controller, and n c is to be determined; since the input signal u(t) and can be directly measured in real time, then the state vector x ψ (t) of the IQC-induced system (5) can also be implemented and calculated.

5. A memory dynamic output feedback control method for an input time-delay automotive suspension according to claim 4, characterized in that, In Step 4, By combining the system (3), the IQC-induced system (5) and the memory dynamic output feedback controller (6), for any \(k\in\{1,2,\cdots,N\}\) λ the closed-loop system is as follows Among them And the relevant system matrix is 6. A memory dynamic output feedback control method for an input time-delay automotive suspension according to claim 5, characterized in that In Step 5, Consider the closed-loop system (7), given scalars and \(q = 1, 2\), if there exist positive definite matrices and scalars such that where Then the closed-loop system (7) with delay d(t) is stable, where and and minimizes the H z 2,max = [1 1] Τ performance index under the output constraint ∞ |u(t)| ≤ u max where the maximum actuator control force constraint in max = (ρ - V(0)) / γ 2 and the disturbance energy is in the range of w 7. The memory dynamic output feedback control method for an input time-delay automotive suspension according to claim 6, further characterized in that In Step 6, Consider the closed-loop system (7), given scalars and \(q = 1, 2\), if there exist positive definite matrices matrices and scalars such that where Then, there exists a dynamic output feedback controller such that the closed-loop system (7) is stable; and the control gain of an ideal dynamic output feedback controller in the form of the memory dynamic output feedback controller (6) is given by the following: and the invertible matrices \(M\) and \(N\) satisfy 8. A memory dynamic output feedback control method for an input time-delay automotive suspension according to claim 7, characterized in that In Step 7, Input parameter r, ρ; The calculation method is as follows a. Solve the following minimization problem b. Mark the optimal solution c. By solving the minimization problem (20); d. Solve the optimal solution Let γ l = γ, l = l + 1; e. When γ l -γ l-1 < -0.1 and t min < 0, repeat steps c - e; otherwise, execute step f; f. End the loop and output the matrix g. Return the controller gain matrices A c , B c0 , B c1 , B c2 , C c .

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