A fault-tolerant controller design method for safety-critical systems

Abstract

The application discloses a fault-tolerant controller design method of a safety-critical system, and comprises the following steps: S1, constructing an actuator partial failure fault model; S2, respectively constructing a Lyapunov function and a control barrier function; S3, respectively constructing a control Lyapunov function constraint condition and a control barrier function constraint condition; S4, constructing a new control barrier function by combining the control barrier function after the constraint condition with fault information; S5, combining the control Lyapunov function and the new control barrier function; S6, solving a control variable by using a quadratic programming method to obtain a fault-tolerant controller of the safety-critical system. The method solves the problem that a general nonlinear affine system is affected by an actuator partial failure fault and further affects the safety and stability of the system, combines the control Lyapunov function constraint condition and the control barrier function constraint condition, calculates a feasible solution for keeping the fault system stable and safe, and obtains an expected control variable.

Description

Technical Field

[0001] This invention relates to the field of fault-tolerant control, specifically to a design method for a fault-tolerant controller for a system with stringent safety requirements. Background Technology

[0002] Two crucial properties of modern nonlinear systems are stability and safety. System stability requires the closed-loop system to eventually reach a predetermined desired state, while system safety requires the closed-loop system to never perform any "bad" actions. Furthermore, if a system's closed-loop trajectory consistently remains within a defined set of states, it is considered safe and does not perform any "bad" actions. Therefore, in modern nonlinear system control problems, both stability and safety must be considered. The premise of designing a system stability controller is that the system must satisfy safety constraints; under the premise of satisfying the safety objective, the goal is to achieve system stability as much as possible.

[0003] For system stability studies, the control Lyapunov function has proven to be a useful tool for designing stabilizing control laws for affine nonlinear systems. The barrier function, a Lyapunov-like function, has applications that can be traced back to optimization problems. The control barrier function is constructed by imitating the extension from the Lyapunov function to the control Lyapunov function. The control barrier function is a means of proving the safety properties of a system based on invariant sets. The barrier function proves the forward invariance of sets without solving for the reachable set of the system state. Research on safety criteria based on the control barrier function mainly studies the behavioral constraints of the system state, i.e., what conditions the barrier function must satisfy to infer that the system state can continuously remain within the safe set.

[0004] In recent years, the Control Barrier Function (CBF) has been applied to many different systems, and the necessary safety controllers have been designed for these systems, such as adaptive cruise control, bipedal robot walking, lane keeping, ship trajectory tracking, and heading control. The Control Barrier Function (CBF) and Control Lyapunov Function (CLF) are part of the constraints in Quadratic Programming (QP). By constraining the objective function through these constraints, QP can calculate the feasible solution of the system under the constraints, which is then used as control input to ensure system stability and safety.

[0005] For general nonlinear systems, existing literature has proposed methods for constructing CBF constraints to ensure system safety. Recently, it has been found that when the system has actuator failures, the CBF constraints obtained using the original safety constraint construction methods cannot satisfy the QP solution to obtain a feasible solution that keeps the system state safe. Summary of the Invention

[0006] To address existing problems, this invention discloses a fault-tolerant controller design method for safety-critical systems. When a general nonlinear system has actuator failures, especially partial failures, the control input obtained by solving the control obstacle function constraint disclosed in this invention can solve the problem of unsafe system states when partial actuator failures occur, ensuring that the faulty system operates within a safe set. In other words, fault-tolerant control is considered on the basis of safety-critical systems.

[0007] A method for designing a fault-tolerant controller for a safety-critical system includes the following steps:

[0008] Step 1: Select the following model of a general nonlinear affine system with actuator failure:

[0009]

[0010] in, Let x(0) be the initial state of the system, and x(t) be the state at time t. It is the control input of the system, ρ is the failure rate of the system when partial failure occurs and 0 < ρ ≤ 1.

[0011] To facilitate the explanation of the design of the control barrier function, system (1) can be further expressed as:

[0012]

[0013] in,

[0014] Step two, select a suitable Lyapunov function and construct a control Lyapunov function (CLF) as a constraint:

[0015] There exists a continuously differentiable function For all system states If there exist positive numbers c1, c2, c3 > 0, then the function V(x) is an exponentially stabilizing controlled Lyapunov function, expressed by the inequality:

[0016] c1||x|| 2 ≤V(x)≤c2||x|| 2 (3)

[0017]

[0018] in,

[0019] Step 3: Based on Step 2, a set of controllers that stabilize the system can be obtained. The set of control variables u can be expressed as:

[0020] Kclf (x)={u∈U:L f V(x)+L g V(x)u+c3V(x)≤0} (5)

[0021] This allows a general nonlinear affine system (1) to have a controller u that satisfies the local Lipschitz condition:

[0022]

[0023] Step four, set the minimum norm controller u * (x) can be considered as a feasible solution to a quadratic programming problem (QP). Using this method, the controlling Lyapunov function can be easily used as a soft constraint for the QP problem, i.e.:

[0024]

[0025] stL f V(x)+L g V(x)u≤-c3V(x)

[0026] Step 5: Select a suitable barrier function and construct a control barrier function (CBF) as the constraint condition:

[0027] Theorem 1: For a general nonlinear affine system (1), there exists a set The set is called a set if and only if for every initial state x(0)∈C, all states t≥0 satisfy x(t)∈C. It is a forward invariant set.

[0028] Theorem 2: Further, we can consider the set Represented as a continuously differentiable function form

[0029]

[0030]

[0031]

[0032] There exists an extended κ-class function α that satisfies the local Lipschitz condition if and only if for all x(t)∈C. It is a barrier function

[0033]

[0034] At this point, set C is a forward-invariant set or a safe set. For all states x(t) ∈ C, if there exists an extended κ-class function α that satisfies the following inequality, then the continuously differentiable function B(x) is the control barrier function:

[0035]

[0036] in At this point, we can consider the set of all control variables that make set C safe as follows:

[0037] K cbf (x)={u∈U:L f B(x)+L g B(x)u+α(B(x))≥0} (13)

[0038] Step six, combining steps two, three, four, and five, yields a single controller that simultaneously satisfies both the control Lyapunov function constraint and the control obstacle function constraint, as follows:

[0039]

[0040]

[0041] in, It is a positive definite matrix. δ is a slack variable. When necessary, the solvability of QP is ensured by relaxing the CLF constraint, that is, relaxing the stability of the system to ensure the safety of the system.

[0042] Step 7: When a partial actuator failure exists, in order to ensure the forward invariance of the safety set C, the inequality (16) of the control barrier function needs to be reconstructed. That is, inequality (16) should include terms that eliminate the effects of partial actuator failure.

[0043] Theorem 3: Combining Theorems 1 and 2, there exists an extended κ-class function α such that for all states x(t) ∈ C and hour, For system (1), when the initial state x(0)∈C, B(x(t))≥0 is always true, and set C is a forward invariant set or a safe set.

[0044] Consider if there exists an extended κ-class function α, when all states x(t)∈C and If the following inequality holds, then the CBFD related to system (1) has been found, ensuring that the system state is in a safe set.

[0045]

[0046] Step 8: Redesign inequality (12) from step 6, and change it to...

[0047] L f B(x)+L g B(x)u+α(B(x))≥F M (x) (18)

[0048] in

[0049] Step nine, reconsider the set of all control variables that make the system safe, and rewrite the inequality (13) in step five as follows:

[0050] K cbfd (x)={u∈U:L f B(x)+L g B(x)uF M (x)+α(B(x))≥0} (19)

[0051] Step 10, the QP expression in step 6 is changed to:

[0052]

[0053] stL f V(x)+L g V(x)u+α(V(x))-δ≤0

[0054] L f B(x)+L g B(x)u+α(B(x))≥F M (x)

[0055] Beneficial effects of the present invention

[0056] The newly designed Control Barrier Function (CBFD) constraint condition is satisfied, demonstrating that the nonlinear system can still meet the system's safety requirements even with partial actuator failure. This solves the problem that the controller cannot guarantee the system remains in a safe state when partial actuator failure occurs.

[0057] To address the impact of actuator partial failure on the safety and stability of general nonlinear affine systems, we propose a novel design approach for existing control barrier function constraints. First, a fault system model is established based on the characteristics of actuator partial failure. Then, the unknown failure rate information is represented in the control barrier function constraints, resulting in the newly designed constraints. Finally, using quadratic programming, the control Lyapunov function constraints and the control barrier function constraints are combined to calculate a feasible solution that maintains the stability and safety of the fault system, i.e., the desired control quantity. The effectiveness of the proposed method is verified through specific case simulations. Attached Figure Description

[0058] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below.

[0059] Figure 1 : Indicates the design process of a fault-tolerant controller for a system with stringent safety requirements.

[0060] Figure 2 When a partial actuator failure occurs in the system, some states enter the unsafe set.

[0061] Figure 3 When a partial actuator failure occurs in the system, the speed of the controlled vehicle...

[0062] Figure 4 When a partial actuator failure occurs in the system, all states remain in the safe set.

[0063] Figure 5 The scaling process of CLF conditions during system simulation.

[0064] Figure 6 When a partial actuator failure occurs in the system, the control input for QP solution is... Detailed Implementation

[0065] The present invention will now be described in further detail with reference to the examples shown in the accompanying drawings.

[0066] In this invention, an adaptive cruise control system is selected as a case study simulation.

[0067] Step 1: Select the following model of a general nonlinear affine system with actuator failure:

[0068]

[0069] in, It represents the state of the system, x = [x1, x2, x3] := [pv] f d],vf v represents the speed of the controlled vehicle. l Let d be the speed of the leading vehicle, d be the distance between the leading vehicle and the controlled vehicle, and p be the distance traveled by the controlled vehicle. Define the initial state of the system as x(0), and the state at time t as x(t). It is the control input of the system, ρ is the failure rate of the system when partial failure occurs and 0 < ρ ≤ 1. m is the weight of the vehicle, Fr(x) = f0 + f1v f +f2v f 2 It is aerodynamic drag, and f0, f1, and f2 are all known constants.

[0070] To facilitate the explanation of the design of the control barrier function, system (21) can be further expressed as:

[0071]

[0072] in,

[0073] Step two, select a suitable Lyapunov function and construct a control Lyapunov function (CLF) as the velocity constraint:

[0074] In this invention, the Lyapunov function selected is V(x) = (v f -v d ) 2 , where v d It is the expected speed that the controlled vehicle can reach.

[0075] There exists a continuously differentiable function For all system states If there exist positive numbers c1, c2, c3 > 0, then the function V(x) is an exponentially stabilizing controlled Lyapunov function, expressed by the inequality:

[0076] c1||x|| 2 ≤V(x)≤c2||x|| 2 (twenty three)

[0077]

[0078] in,

[0079] Step 3: Based on Step 2, a set of controllers that stabilize the system can be obtained. The set of control variables u can be expressed as:

[0080] K clf (x)={u∈U:L f V(x)+L gV(x)u+c3V(x)≤0} (25)

[0081] This allows a general nonlinear affine system (1) to have a controller u that satisfies the local Lipschitz condition:

[0082]

[0083] Step four, set the minimum norm controller u * (x) can be considered as a feasible solution to a quadratic programming problem (QP). Using this method, the controlling Lyapunov function can be easily used as a soft constraint for the QP problem, i.e.:

[0084]

[0085] stL f V(x)+L g V(x)u≤-c3V(x)

[0086] Step 5: Select a suitable obstacle function and construct a control obstacle function (CBF) as a safety distance constraint:

[0087] When selecting the obstacle function, the displacement of the controlled vehicle under maximum braking conditions needs to be considered, as this directly affects the safe distance between the two vehicles. The input of the control variable must satisfy the inequality -a d mg≤u≤a i mg, where g is the gravitational acceleration constant, a d and a i Let g be the proportionality coefficient for deceleration and acceleration. At this point, we have:

[0088]

[0089] Wherein, time constant d(t) is the distance at the current moment, and the distance after T seconds is...

[0090] At this point, the obstacle function serving as the safety distance constraint is:

[0091]

[0092] For all states x(t)∈C, if there exists an extended κ-class function α that satisfies the following inequality, then a continuously differentiable function B(x) is a control barrier function:

[0093]

[0094] in At this point, we can consider the set of all control variables that make set C safe as follows:

[0095] K cbf (x)={u∈U:L f B(x)+L g B(x)u+α(B(x))≥0} (32)

[0096] Step six, combining steps two, three, four, and five, we can derive a single controller that simultaneously satisfies both the control Lyapunov function constraint and the control obstacle function constraint in the following way:

[0097]

[0098] stL f V(x)+L g V(x)u+α(V(x))-δ≤0

[0099] L f B(x)+L g B(x)u+α(B(x))≥0

[0100] in, It is a positive definite matrix. δ is a slack variable. When necessary, the solvability of QP is ensured by relaxing the CLF constraint, that is, relaxing the stability of the system to ensure the safety of the system.

[0101] In this step, the QP objective function is proposed based on the performance objective of achieving the CLF constraints. The purpose of the soft constraints is to ensure that the controlled vehicle reaches the set desired speed. However, to achieve the desired speed, the slack variables must be balanced to ensure the solvability and continuity of the QP. The CLF first uses feedback u = F r +mμ partially linearizes the system. Ultimately, the cost function related to the control input can be chosen as μ. T μ is represented as:

[0102]

[0103] This allows us to determine H in the QP objective function. acc and F acc That is

[0104] Where δ s It is the weighting coefficient of the slack variable δ.

[0105] Step 7: When a partial actuator failure exists, to ensure the forward invariance of the safety set C, the inequality (33) of the control barrier function needs to be reconstructed. That is, inequality (33) should include terms related to eliminating the effects of partial actuator failure. Consider if there exists an extended κ-class function α, where all states x(t) ∈ C and If the following inequality holds, then the CBFD related to system (21) has been found, ensuring that the system state is in a safe set.

[0106]

[0107] Step 8: Redesign inequality (33) from step 6, and change it to...

[0108] L f B(x)+L g B(x)u+α(B(x))≥F M (x) (36)

[0109] in

[0110] Step nine, reconsider the set of all control variables that make the system safe, and rewrite the inequality (32) in step five as follows:

[0111] K cbfd (x)={u∈U:L f B(x)+L g B(x)uF M (x)+α(B(x))≥0} (37)

[0112] Step 10, the QP expression in step 6 is changed to:

[0113]

[0114] stL f V(x)+L g V(x)u+α(V(x))-δ≤0

[0115] L f B(x)+L g B(x)u+α(B(x))≥F M (x)

[0116] A simulation example is used to verify the effectiveness of the control barrier function design method for actuator partial failure in general nonlinear systems:

[0117] In the simulation examples of this invention, the values ​​of the known constants are as follows. The gravitational acceleration g is taken as 9.81 m / s². 2 Mass m = 1650 kg, constants f0 = 0.1, f1 = 5, f2 = 0.25, desired velocity v d =24m / s, acceleration / deceleration ratio coefficient a d =a i =0.4,

[0118] In addition, the range of actuator failure coefficient was set in the simulation as follows: Let be the lower and upper bounds of ρ, and

[0119] Further determine F M The value of (x), Since u∈[u min ,u max ],-a d mg≤u≤a i mg, v f ∈[v fmin ,v fmax ],0≤v f ≤v d Then we can easily obtain the following inequality:

[0120]

[0121] according to Figure 2 It can be clearly seen that... ρ =0.2, If the system has partial failures, and no action is taken to address the system's control obstacle function constraints, the system state will exceed the safe set and enter an unsafe state.

[0122] Figure 3 This means that initially, the CBFD value is relatively large, indicating a sufficiently large safety distance. At this point, the system will gradually accelerate to reach the set desired speed. When the CBFD value approaches 0, the system can only change the value of the slack variable, such as... Figure 5 When the soft constraints are relaxed, the speed of the controlled vehicle begins to decrease.

[0123] according to Figure 4 It's easy to see that B(x) ≥ 0 is always satisfied. According to Theorem 3, this means the system state is always in the safe set. (Compare...) Figure 2 and Figure 4 It can be easily concluded that in the proposed new method, constructing CBFD introduces F M (x) can eliminate the impact of actuator partial failure on system safety, ensuring that the system state remains within the safe set even under fault conditions. This guarantees the stable and safe operation of the aforementioned faulty system.

[0124] Figure 5 , represents the scaling process of CLF in order to satisfy the hard constraint condition, that is, the CBFD constraint condition of the safety distance.

[0125] Figure 6The control input obtained by QP solution satisfies the CLF and CBFD constraints, which can ensure the safe and stable operation of the system.

[0126] This invention studies a fault-tolerant controller design method for safety-critical systems. To address the issue of actuator partial failure affecting the safety and stability of general nonlinear affine systems, a novel design approach is proposed for existing control barrier function constraints. First, a fault system model is established based on the characteristics of actuator partial failure. Then, the unknown failure rate information is represented in the control barrier function constraints, resulting in the newly designed constraints. Finally, using quadratic programming, the control Lyapunov function constraints and the control barrier function constraints are combined to calculate a feasible solution that maintains the stability and safety of the fault system, thus yielding the desired control quantity. The effectiveness of the proposed method is verified through specific case simulations.

Claims

1. A method for designing a fault-tolerant controller for a safety-critical system, characterized in that, Includes the following steps: S1. Construct a failure model for the actuator section; The method for establishing the actuator failure model in S1: S1-1. Establish a model for a general nonlinear affine system with actuator failure: S1-2. Based on the established model of actuator failure in a general nonlinear affine system, the fault model is rewritten to obtain the fault model. in, , , It is the state of the system, in The state at time is , It is the system's control input. The failure rate of the system is the rate at which partial failure occurs. ; S2. Construct the Lyapunov function and the control barrier function respectively; The design method for the control barrier function is as follows: For general nonlinear affine systems, there exists a set If and only if every initial state For all All conditions are met ,gather It is a forward-invariant set; set Represented as a continuously differentiable function form There exists an extension that satisfies the local Lipschitz condition. Class function If and only if all hour, It is a barrier function ， At this time, the set It is a forward-invariant set or a safe set for all states. If an extension exists Class function If the following inequalities are satisfied, then a continuously differentiable function is... It is a control barrier function: ； S3. Construct the constraints for controlling the Lyapunov function and the constraints for controlling the obstacle function, respectively. The constraints of the control barrier function include terms related to eliminating the effects of faults: S4. Construct a new control obstacle function by combining the control obstacle function with the fault information after constructing the constraint conditions. In step S4, a new control barrier function is constructed by combining the fault information: in , , , The set of control variables for system safety is obtained based on the control barrier function; ； S5. Combining the control Lyapunov function with a new control barrier function; S6. Use quadratic programming to solve for the control variables and obtain the fault-tolerant controller for the safety-critical system.

2. The fault-tolerant controller design method for a safety-critical system according to claim 1, characterized in that, In step S3, the design method for controlling the Lyapunov function is as follows: There exists a continuously differentiable function For all system states If a positive number exists Then the function It is an exponentially stabilizing control Lyapunov function, expressed by the inequality as follows: in, , , A set of controllers is obtained to stabilize the system, and the control quantity is... The set can be represented as: This would allow for the existence of a controller in a general nonlinear affine system that satisfies the local Lipschitz condition. for: 。 3. The fault-tolerant controller design method for a safety-critical system according to claim 2, characterized in that, In step S6, a single controller that simultaneously satisfies the control Lyapunov function constraint and the control obstacle function constraint is obtained through quadratic programming. in, , , It is a positive definite matrix. , It is a slack variable.

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