High-precision preset performance controller design method under unified boundary constraint

Abstract

The invention discloses a high-precision preset performance controller design method under a unified boundary constraint, and the method comprises the following steps: 1), designing a precise preset performance boundary capable of automatically matching an initial constraint condition through introducing a proper adjustment function; 2) providing a high-precision preset performance control algorithm under precise boundary constraint for different error initial values; 3) by establishing a proper scale transformation function, designing a unified preset performance boundary capable of presenting various constraint forms under different parameter selections; and 4) providing a high-precision preset performance control algorithm under the unified boundary constraint for different initial conditions. According to the method, the tracking error can be converged to the given precision within the preset time with extremely small overshoot under different initial value conditions. In addition, according to the method, various performance constraints can be realized under a fixed control structure, and the tracking error can be converged to the preset precision in the preset time under different initial conditions.

Description

Technical Field

[0001] The present invention relates to a method for designing a controller, and particularly to a method for designing a high-precision preset performance controller under unified boundary constraints. Background Art

[0002] With the rapid development of science and technology, people's requirements for the control performance of controlled systems are getting higher and higher. In actual industrial production, the vast majority of systems are essentially highly complex and uncertain nonlinear systems. At present, there have been many research results on control algorithms for single-input single-output (SISO) nonlinear systems. The research on the high-performance control problem of multi-inputs multi-outputs (MIMO) nonlinear systems is relatively less. And many actual engineering application systems are mostly MIMO systems, such as six-degree-of-freedom robotic arms and quadrotor UAVs widely used in various industries. Therefore, designing a suitable control algorithm for such MIMO nonlinear systems to execute various tasks efficiently and accurately is the goal that people have been pursuing.

[0003] Now, researchers have gradually realized that combining transient performance and steady-state performance can more comprehensively evaluate the effectiveness of control algorithms. Taking the common six-degree-of-freedom robotic arm as an example, during operation, it will be affected by model uncertainties and external disturbances, and the efficiency of the robotic arm to complete tasks is directly related to its transient performance. The efficiency of the robotic arm to complete tasks is related to the convergence time of tracking a pre-given desired trajectory. The smaller the convergence time, the faster the robotic arm completes tasks and the higher the efficiency. At the same time, how to ensure that the robotic arm completes tasks smoothly and accurately is directly related to the overshoot and steady-state error of the tracking error. Therefore, how to ensure the realization of these performance indicators in controller design is very crucial. In actual engineering applications, it is often expected that the controlled system has a sufficiently short convergence time, sufficiently high tracking accuracy, and sufficiently small overshoot. To achieve these performance indicators, many scholars are committed to studying how to design a suitable controller to improve the steady-state and transient performance of the system. In solving the preset tracking performance problem, preset performance control is one of the current mainstream control methods and is widely used to achieve high-performance control of various systems.

[0004] Consider a class of uncertain MIMO nonlinear normal form systems:

[0005]

[0006] Wherein, and are measurable system states, is an unknown time-varying nonlinear function, is a time-varying unknown asymmetric control gain matrix, is an unknown parameter vector, and are the control input and system output respectively, is an unknown bounded external disturbance. In addition, define as the tracking error, where is the desired trajectory. Without causing ambiguity, the independent variable parameters of some functions will be omitted later, such as: is abbreviated as f.

[0007] There are many practical systems that belong to this form of nonlinear system, such as: Euler-Lagrange systems such as robots. Taking the m-degree-of-freedom manipulator system as an example, its dynamic model is:

[0008]

[0009] where, and are the joint position, velocity and acceleration of the manipulator respectively, and are the inertia matrix, Coriolis force and centripetal force matrix and gravity vector respectively, and the vector and are the joint friction, control input and external disturbance respectively.

[0010] If the following two state variables are defined: x1 = q and then the manipulator dynamic model (2) becomes:

[0011]

[0012] Obviously, compared with the second-order MIMO multi-degree-of-freedom manipulator system, the high-order canonical form MIMO nonlinear system shown in (1) is more general. Therefore, it can be considered that Euler-Lagrange systems such as robots are a special case of the system studied in the present invention, so the control algorithms designed in the present invention generally also apply to practical systems such as manipulators, drones, and unmanned vehicles.

[0013] For the high-precision preset performance control problem of system (1), the existing research results have the following limitations:

[0014] (i) The multiplication commutative law in scalars generally does not hold for matrix multiplication, and an additional diagonal performance matrix under the performance constraint will appear in front of the original control gain matrix, changing its structure, which makes the controller design and stability analysis challenging.

[0015] (ii) The error feasible domain under the classical performance boundary is relatively large, which can easily cause jitter or overshoot in the control process. In addition, the preset performance control must meet the initial performance constraints, but the design parameters of the existing preset performance functions are mostly fixed constants. Once the expected trajectory changes or the system is restarted, the initial conditions need to be rechecked to avoid control singularities.

[0016] (iii) The precise initial values of the system may be difficult to obtain in some special environments (deep sea, strong wind, etc.), but most of the existing preset performance control schemes that do not depend on initial conditions can generally only achieve a specific form of performance constraint under a fixed control structure, which means that the control algorithm lacks versatility. Summary of the invention

[0017] In view of the above-mentioned deficiencies in the prior art, the present invention provides a design method for a high-precision preset performance controller under unified boundary constraints.

[0018] A high-precision preset performance controller design method under unified boundary constraints, the method comprising the following steps:

[0019] 1) By introducing appropriate adjustment functions, precise preset performance boundaries are designed that can automatically match the initial constraints;

[0020] 2) For different initial error values, a high-precision preset performance control algorithm under precise boundary constraints is proposed;

[0021] 3) By establishing a suitable scale transformation function, a unified preset performance boundary is designed that can present various constraint forms under different parameter selections;

[0022] 4) For different initial conditions, a high-precision preset performance control algorithm under unified boundary constraints is proposed.

[0023] Compared with the prior art, the present invention has the following technical effects:

[0024] 1) The present invention designs a high-precision preset performance control algorithm under precise boundary constraints, which can not only make the tracking error converge to a given accuracy with extremely small overshoot within a preset time, but also realize automatic matching of initial performance constraints under the conditions of expected trajectory change or system restart, avoiding the time-consuming and labor-intensive process of manually matching the constraints of each dimension.

[0025] 2) The present invention designs a high-precision preset performance control algorithm under unified boundary constraints, which can achieve multiple performance constraints under a fixed control structure and can also make the tracking error converge to a predetermined accuracy at a preset time under different initial conditions, greatly improving the versatility of the designed control algorithm. In addition, this algorithm can impose a more stringent constraint mechanism on the dynamic characteristics of the error, and can significantly improve the transient and steady-state response characteristics under the same operating conditions. Description of the Drawings

[0026] Figure 1 is the operating curve graph of the desired trajectory and the actual output of the system;

[0027] Figure 2 is the operating curve graph of the tracking error;

[0028] Figure 3 is the operating curve graph of the control input;

[0029] Figure 4 is the operating curve graph of the virtual parameter;

[0030] Figure 5 is the operating curve graph of the desired trajectory and the trajectory tracking;

[0031] Figure 6 is the operating curve graph of the tracking error;

[0032] Figure 7 is the operating curve graph of the control input;

[0033] Figure 8 is the operating curve graph of the virtual parameter;

[0034] Figure 9 is the operating curve graph of the desired trajectory and the actual output;

[0035] Figure 10 is the operating curve graph of the tracking error;

[0036] Figure 11 is the operating curve graph of the control input;

[0037] Figure 12 is the operating curve graph of the virtual parameter;

[0038] Figure 13 is the operating curve graph of the desired trajectory and the actual output;

[0039] Figure 14 is the operating curve graph of the tracking error;

[0040] Figure 15 is the operating curve graph of the control input;

[0041] Figure 16 is the operating curve graph of the virtual parameter. Detailed implementation manners

[0042] The present invention will be further described in detail below with reference to the accompanying drawings and specific implementation manners.

[0043] A high-precision pre-set performance controller design method under unified boundary constraints, the control objective of which is to design a suitable pre-set performance controller such that: For a time-varying asymmetric control gain matrix, all signals in the closed-loop system (1) are bounded under the condition of considering performance constraints; The tracking error converges to a given accuracy within a preset time with a sufficiently small overshoot, and even if the desired trajectory changes or the system restarts, the initial performance constraint conditions can be automatically matched; Multiple performance constraints can be achieved under a fixed control structure, and the tracking error can converge to a predetermined accuracy within a preset time under different initial conditions.

[0044] In order to achieve the above control objectives, the following assumptions are given.

[0045] Assumption 1 The desired trajectory y d and its up to nth-order derivatives are known to be bounded and piecewise continuous.

[0046] Assumption 2 According to the rough structural information of the unknown nonlinear functions and it is known that there exists a non-negative unknown constant a≥0 and a known smooth function such that holds. In addition, if is bounded, then is bounded.

[0047] Assumption 3 For the control gain matrix there exists an unknown but positive definite diagonal matrix such that is uniformly positive definite or negative definite. Without loss of generality, the present invention assumes that is uniformly positive definite. In addition, there exist some unknown constants m1, m2 and g , such that for any variable there is ||M||≤m1, and hold.

[0048] Assumption 1 is adopted by most controller designs when designing a controller for the system (1). Assumption 2 means that according to only the rough structural information, from the non-parametrizable nonlinear functions and "Core" information can be obtained, making the designed controller structure simpler and more computationally efficient. Assumption 3 can ensure that the system (1) is controllable. By using the auxiliary matrix M, the present invention has relatively weaker restrictions on the control gain matrix. For the robot system (3), since the control gain term H -1 is a symmetric positive definite matrix, the auxiliary matrix M in Assumption 3 is the identity matrix I. Therefore, Assumption 3 is more general in its assumptions about the control gain matrix G.

[0049] A high-precision preset performance controller design method under unified boundary constraints, the method comprising the following steps:

[0050] 1) High-precision preset performance control under precise boundary constraints

[0051] 1.1) Precise preset performance boundary

[0052] To make the tracking error enter the preset steady-state accuracy within a preset time, a finite-time performance function ρ j (t) is adopted. Such functions have the following properties: 1) ρ j (0) = 1 and ρ j (t) = ρ j,T for t ≥ T, where T > 0 is the convergence time that can be arbitrarily specified in advance by the user; 2) 3) is continuous for t ≥ 0; The finite-time performance function satisfying the above properties is designed as follows:

[0053]

[0054] where ρ j,T and b1 are design parameters, ρ j,T > 0 is the maximum allowable steady-state error, and b1 ≥ 1 is used to adjust the convergence rate of the function ρ j (t);

[0055] To improve the accuracy constraint of the finite-time performance function (4), the present invention introduces a suitable adjustment function Such functions have the following properties: 1) is a strictly decreasing function on [0, t); 2) and for t ≥ T, where T > 0 is the user-defined convergence time; 3) always holds for t ≥ 0. The adjustment function satisfying the above properties can be designed as follows:

[0056]

[0057] where b2 is a design parameter, and b2 ≥ 1 is used to adjust the function The convergence rate. It is not difficult to see that for \(0\leq t < T\), the function The common form of etc.

[0058] According to (4) and (5), the present invention designs the precise preset performance boundary as follows:

[0059]

[0060] Wherein, and σ j are design parameters, and are used to automatically match different initial error values, and σ j > 0 are the upper and lower bound constants related to the tracking accuracy respectively.

[0061] When the precise preset performance boundary is designed, it is used to constrain the tracking error as follows:

[0062]

[0063] Lemma 1 For different initial error values \(e j (0)\), if \(e j (t)\) satisfies the performance constraint inequality (7), then the following properties hold: 1) \(e j (t)\) enters the preset accuracy within the preset time; 2) The overshoot of \(e j (t)\) during the tracking process can be small enough; 3) The initial performance constraint condition can be automatically matched.

[0064] Proof: 1) When \(t\geq T\), there is and \(\rho j (t)\equiv\rho j,T \), then φ j (t)\equiv σ j \rho j,T and Thus, there is That is, the tracking error \(e j (t)\) enters the preset accuracy within the preset time \(T\) ( σ j \rho j,T , σ j \rho j,T );

[0065] 2) For the convenience of analysis, assume the constant parameter and Then Since and ρ j (t) ≤ 1, then When is small enough, the upper and lower boundary distances of the exact performance are very small, and the error feasible region is very small, so that the overshoot is small enough;

[0066] 3) Since the adjustment function design parameter σ j > 0 and there is always holds, then the exact preset performance boundary can flexibly adjust itself to match any initial error value.

[0067] 1.2) Controller design and stability analysis

[0068] It is difficult to directly design a controller to make e j (t) satisfy the performance constraint (7). The transformation error is introduced as follows:

[0069]

[0070] where It can be proved that by using (8), the original constrained problem can be transformed into an equivalent unconstrained problem, that is: the constraint problem that e j (t) varies strictly within the performance boundary can be simplified to ensure the stability problem that ζ j (t) is bounded.

[0071] Next, differentiating ζ j (t) gives:

[0072]

[0073] where

[0074]

[0075] To facilitate controller design, the filtering variable s j is introduced as follows:

[0076]

[0077] where β j is a design parameter.

[0078] According to (9), using mathematical induction, it can be further obtained that:

[0079]

[0080] According to (10) and (11), for s jTaking the derivative, we get:

[0081]

[0082] Furthermore, (12) can be written in a compact matrix form as follows:

[0083]

[0084] where and and the function v j is

[0085] Since e = x1 - y d , then according to (1), it can be deduced that:

[0086]

[0087] According to (13) and (14), it can be obtained that:

[0088]

[0089] where is a known function.

[0090] Select the first Lyapunov candidate function as From (15), its derivative is:

[0091]

[0092] where

[0093] According to Assumption 2 and Assumption 3, using the Young inequality, we have:

[0094]

[0095] Also, since then it can be obtained that:

[0096]

[0097] where is an unknown virtual parameter, is a scalar function that can be used for controller design.

[0098] Thus, (16) can be simplified to:

[0099]

[0100] Select the second Lyapunov candidate function as Then its derivative is:

[0101]

[0102] wherein, is an unknown constant, λ2>0 is a design parameter, is the estimation error of the virtual parameter.

[0103] Finally, the total Lyapunov candidate function is selected as:

[0104]

[0105] Therefore, can be expressed as:

[0106]

[0107] Next, the robust adaptive controller is designed as follows:

[0108]

[0109] wherein, c1>0, λ1>0 and λ2>0 are design parameters, ψ1 is the core function related to the modeling uncertainty, is the estimated value of the unknown virtual parameter a1, is an arbitrarily selectable initial value. Since λ2ψ1||s|| 2 ≥0, it is not difficult to obtain that for the initial value always holds.

[0110] According to the designed robust adaptive controller (24), the following theorem can be obtained.

[0111] Theorem 1 For the nonlinear system (1), under the conditions of satisfying Assumptions 1 - 3, according to the robust adaptive controller (24), there exist appropriate design parameters such that the control objectives and can be achieved.

[0112] Proof: Substitute the control law u of (24) into (23), and we can get:

[0113]

[0114] Since both M and R are positive definite diagonal matrices, then MR = RM. Also, because 2MG = (MG + G T M) + (MG - G T M), where (MG + G T M) is a symmetric matrix and (MG - G T M) is an anti-symmetric matrix, then:

[0115]

[0116] Among them, the

[0117] From assumption 3, it can be obtained that:

[0118]

[0119] Among them, λ R is the lower bound value of λ min {R T R}.

[0120] Let Then there is:

[0121]

[0122] Substituting (28) into (25), it can be further obtained that:

[0123]

[0124] Substituting the adaptive update rate of (24) into (29), it can be obtained that:

[0125]

[0126] Since then

[0127]

[0128] Also because Among them is the upper bound value of λ max {M}, then Then there is:

[0129]

[0130] Among them,

[0131] Integrating both sides of the inequality (32) simultaneously, it can be obtained that:

[0132]

[0133] It can be further obtained the following conclusion: (1) For any finite initial value, obviously V1(0) is bounded. According to (33), then V1(t) is bounded. From (22), it can be known that s and are also bounded, so that s j is bounded. Also because s j is bounded, then from (11), it can be known that ζ j is also bounded. Further from (8), it can be known that Πj (t) and e j is also bounded. Also, since e j = x 1j - y dj , and the reference trajectory y dj is bounded, the system state x 1j (x1) is bounded. Through similar analysis steps, it can be known that the system states x2, …, x n , the matrix R and the vector v are all bounded. Thus, from (24), it can be seen that the control input u and its adaptive update rate are also bounded. In summary, all signals in the closed-loop system are bounded. (2) Since ζ j is bounded, it can be known that always holds for t ∈ [0, +∞). Further, according to Lemma 1, it can be known that the tracking error converges to the given accuracy with a sufficiently small overshoot within a preset time. In addition, even if the desired trajectory changes or the system restarts, the initial performance constraint conditions can be automatically matched.

[0134] 2) High-precision preset performance control under unified boundary constraints

[0135] 2.1) Unified preset performance boundary

[0136] First, construct the scaling function as follows:

[0137]

[0138] where q j ∈ {0, 1} is a design parameter, b3 ≥ 1 is used to determine the convergence rate of the function κ(t), κ(t) is a strictly increasing function and satisfies κ(0) = 0 and κ(T) = 1, T > 0 is the user-defined convergence time. Then, the present invention selects κ(t) = sin[(π / 2)(t / T)]. Other common forms of κ(t) are 1 - (1 - t / T) 2 , cos[(π / 2)(t / T - 1)], etc. It is not difficult to see that the scaling function η j (t) has the following properties: 1) When q j = 1, η j (t) satisfies η j (t) ≡ 1 on [0, +∞); 2) When q j = 0, η j (t) is monotonically increasing on [0, T), satisfies η j (0) = 0 and η j (t) = 1 when t ≥ T; 3) No matter what value q j takes, η j (t) is continuously differentiable and bounded, and its first derivative is also bounded.

[0139] In order to decouple the constraint bound from the tracking error, the transfer error is introduced as follows:

[0140] χ j (t) = η j (t)e j (t) (35)

[0141] Using the defined transfer error χ j (t), 1.1) The exact preset performance boundary of the design becomes:

[0142]

[0143] In order to achieve various performance constraints under a fixed control structure, the present invention designs a unified preset performance boundary as follows:

[0144]

[0145] in, η j (t) and is the scaling function defined by (34), p j , q j and is the design parameter. It is worth noting that a small positive constant can be Embedded Scaling Function In, so when and Sometimes Otherwise, the boundary is infinite, which leads to singular problems at the initial moment.

[0146] After designing a unified preset performance boundary, it is used to constrain the tracking error as follows:

[0147]

[0148] Lemma 2 For different initial conditions (e j (0) precisely known, known direction, and unknown direction), if the tracking error e j (t) satisfies the performance constraint inequality (38), then the following properties hold: 1) Under different parameter selections, e j (t) performance constraints take many forms; 2)e j (t) Converge to the predetermined accuracy within the pre-set time.

[0149] Proof: 1) By selecting the key design parameter q j , p j, q j and (38) will exhibit different constraint forms: (i) If the tracking error is precisely known (e j (0) = e j0 (e j0 is a known real number)), the key design parameters are selected as q j = 1, q j = 1, p j = 1 and (38) becomes According to (36), it can be known that This means that there are both upper and lower bounds on the initial error, and better tracking performance can be achieved under precise feasible region constraints. (ii) If the direction of the tracking error is known (e j (0) < 0), the key design parameters are selected as q j = 0, q j = 0, p j = 1 and (38) becomes Note that then Also, because and Thus, -∞ < e j (0) < 0, which means that there is an upper bound on the initial error but no lower bound, and it is applicable to any scenario where sign(e j (0)) = -1, and the exact value of the initial error is no longer required. (iii) If the direction of the tracking error is known (e j (0) > 0), the key design parameters are selected as q j = 0, q j = 0, p j = 0 and (38) becomes Based on the analysis in (ii), it is easy to obtain 0 < e j (0) < +∞, which means that there is no upper bound on the initial error but there is a lower bound, and it is applicable to any scenario where sign(e j (0)) = 1, and the exact value of the initial error is also not required. (iv) If the direction of the tracking error is unknown (e j (0) > 0 or e j (0) < 0), the key design parameters are selected as q j= 0, q j = 0, p j = 0 and (38) becomes According to the analysis in (ii), it is easy to obtain that -∞ < e j (0) < +∞, which means that there is neither an upper bound constraint nor a lower bound constraint on the initial error. Noting that the performance boundary at this time has nothing to do with the initial error, a global control scheme can be obtained, that is, the designed controller does not depend on the system initial value and is applicable to any initial condition. However, it is necessary to select appropriate controller design parameters to reduce overshoot.

[0150] 2) When t ≥ T, there is ρ j (t) ≡ ρ j,T and η j (t) ≡ 1, then and Thus, cases (i)-(iv) in 1) all become That is: under different initial conditions (e j (0) is exactly known, the direction is known, and the direction is unknown), the tracking error e j (t) enters the preset accuracy within the preset time T

[0151] 2.2) Controller design and stability analysis

[0152] When the design parameters q j , p j , q j and take different numerical values, the unified preset performance boundary (37) will show various constraint forms under different design parameters:

[0153] ① is always positive and is positive and negative at times;

[0154] ② is always negative and is negative and positive at times;

[0155] ③

[0156] ④

[0157] ⑤

[0158] To equivalently transform the performance constraint problem (38) into an unconstrained bounded problem, the present invention designs a special non - linear transformation function as follows:

[0159]

[0160] Wherein, τ j > 0 is a design parameter.

[0161] Before designing the controller, the following useful lemma is given, which can prove that by using the above non - linear transformation function (39), the original constrained problem can be transformed into an equivalent unconstrained problem, that is: the tracking error e j (t) strictly varying within the performance boundary can be simplified to a stability problem of ensuring that ε j (t) is bounded.

[0162] Lemma 3: For any initial condition satisfying , if ε j (t) is bounded, then the inequality constraint always holds for t ∈ [0, + ∞).

[0163] Proof: When t = t1, assume or Because and e j (t) is a continuous function, according to the intermediate value theorem, there exists a certain moment 0 < t2 < t1 such that or Thus, ε j (t) = + ∞, which contradicts the boundedness of ε j (t). Therefore, always holds for t ∈ [0, + ∞).

[0164] According to (39), for ε j (t), we have:

[0165]

[0166] Wherein,

[0167]

[0168] For the sake of convenience of description, (40) is written in a compact matrix form as follows:

[0169]

[0170] Wherein, and

[0171] According to (41), by using mathematical induction, it can be obtained that:

[0172]

[0173] For the convenience of controller design, the filtering variable ξ is introduced as follows:

[0174]

[0175] where β1, β2, …, β n-1 are design parameters, and their selection needs to satisfy a n-1 + β n-1 a n-2 + … + β1 to be a Hurwitz polynomial.

[0176] According to (42) and (43), by taking the derivative of ξ, it can be obtained that:

[0177]

[0178] Since e = x1 - y d , then according to (1), it can be deduced that:

[0179]

[0180] According to (45), then (44) can be further transformed into:

[0181]

[0182] where the computable function is:

[0183]

[0184] Select the first Lyapunov candidate function as It can be known from (46) that its derivative is:

[0185]

[0186] where

[0187] According to Assumption 2 and Assumption 3, by using (triangle inequality), it can be obtained that:

[0188]

[0189] where is an unknown virtual parameter, is a scalar function that can be used for controller design.

[0190] Thus, (47) can be simplified to:

[0191]

[0192] Select the second Lyapunov candidate function as Then its derivative is:[[]]

[0193]

[0194] Where,[[]] is an unknown constant, λ2>0 is a design parameter,[[]] is the estimation error of the virtual parameter.[[]]

[0195] Finally, select the total Lyapunov candidate function as:[[]]

[0196]

[0197] Therefore,[[]] can be expressed as:[[]]

[0198]

[0199] Next, design the robust adaptive controller as follows:[[]]

[0200]

[0201] Where, c2>0, λ3>0, λ4>0 and[[]] are design parameters,[[]] is the estimated value of the unknown virtual parameter a2 (related to the modeling uncertainty),[[]] is an arbitrarily selectable[[]] initial value of. Because[[]] Then it is not difficult to obtain that for the initial value[[]] always holds.[[]]

[0202] According to the designed robust adaptive controller (53), the following theorem can be obtained.[[]]

[0203] Theorem 2 For the nonlinear system (1), under the conditions of satisfying Assumptions 1-3, according to the robust adaptive controller (53), there exist appropriate design parameters such that the control objectives[[]] and[[]] can be achieved.[[]]

[0204] Proof: Substitute the control law u of (53) into (52) to get:[[]]

[0205]

[0206] Similar to the analysis in Step 1.2), it is easy to obtain:[[]]

[0207]

[0208] According to Hypothesis 3, we have:

[0209]

[0210] According to (55) and (56), we have:

[0211]

[0212] where

[0213] Thus, the derivative of v2 in (54) can be further simplified to:

[0214]

[0215] where the following is used

[0216] Next, substituting the adaptive update rate in (53) into (58), we get:

[0217]

[0218] It can be seen that where is the upper bound value of λ max for {M}, then Since then (59) becomes:

[0219]

[0220] where

[0221] By integrating both sides of (60), we get:

[0222]

[0223] Therefore, it is easy to obtain: (1) For any finite initial value, V2(0) is bounded. According to (61), then V2(t) is bounded. From (51), it can be seen that ξ and are also bounded. Also, since ξ is bounded, then from (43), it can be seen that ε is bounded. According to (39) and (40), it can be known that the error signal e, the matrix Γ, and the vector w are all bounded. And because e = x1 - y d , and the reference trajectory y d is bounded, then the system state x1 is bounded. Similarly, it can be known that the system states x2,..., x n , the matrix Γ (k) and the vector w (k) (k = 1, 2,..., n - 1) are all bounded. Thus, from (53), it can be seen that the control input u and its adaptive update rate is also bounded. In summary, all signals in a closed-loop system are bounded. (2) Since ε(ε j ) is always bounded. According to Lemma 3, It always holds true on t∈[0,+∞). Furthermore, according to Lemma 2, under different parameter selections, e j The performance constraints of (t) are presented in various forms, that is, various performance constraints can be achieved under a fixed control structure. In addition, under different initial conditions, e j (t) It can also converge to the predetermined accuracy within the time set in advance.

[0224] 3) Algorithm simulation and result analysis

[0225] 3.1 Tracking control under precise boundary constraints

[0226] To verify the effectiveness of the control algorithm in step 1.2), consider the MIMO nonlinear standard system as follows:

[0227]

[0228] in,

[0229]

[0230] Among them, θ 11 =θ 12 =θ 21 =0.1. Obviously, G and G+G T Neither positive definite nor negative definite. However, there exists an auxiliary matrix M that makes assumption 3.3 of the present invention valid, where M is:

[0231]

[0232] Simulation 1 (Tracking control under precise preset performance boundary): In this simulation, the desired trajectory is set to y d =[0.2cos(0.2πt),0.2sin(0.2πt)] T The initial state of the system is x1(0)=[1.2,-1] T , x2(0)=[0,0] T and The relevant parameter for the exact performance boundary is chosen as ρ 1,T =ρ 2,T =0.2, σ 1=0.1, σ 2 = 0.2, b1=b2=b3=3 and T=3. The controller parameters are selected as c1=0.02, λ1=1, λ2=0.0001 and β=1. The simulation results are shown in Figures 1 - 4 shown.

[0233] From Figure 2 It can be seen that the tracking error converges to the given precision e1(t) ∈ (-0.02, 0.04), t ≥ 3s and e2(t) ∈ (-0.04, 0.02), t ≥ 3s with a preset convergence time T = 3s, and the overshoot of the tracking process is small enough. Figure 1 is the running curve graph of the desired trajectory and the actual output of the system, Figure 3 is the running curve graph of the control input, Figure 4 is the running curve graph of the virtual parameter. It can be seen that all these signals are bounded, which is consistent with the above-mentioned system stability analysis results.

[0234] Simulation 2 (automatic matching of initial performance constraint conditions under system restart): If the system restarts due to a fault, the current initial state is obviously inconsistent with the set value in Simulation 1. Assume that the current initial state of the system becomes: x1(0) = [-0.6, 0.8] t , x2(0) = [0, 0] T and The remaining settings are the same as in Simulation 1. The simulation results of the tracking error are as Figures 5 - 8 shown.

[0235] From Figure 6 it can be seen that if the initial state of the system changes (due to system restart), without readjusting the performance boundary and controller design parameters (i.e., the relevant design parameters in Simulation 1 and Simulation 2 are the same), the tracking errors e1 and e2 still strictly vary within the designed performance boundary, that is: for any initial value of the tracking error, the designed precise preset performance boundary can automatically match the initial performance constraint conditions. Figure 5 is the running curve graph of the desired trajectory and the actual output of the system, Figure 7 is the running curve graph of the control input, Figure 8 is the running curve graph of the virtual parameter. It can be seen that all these signals are bounded, which is consistent with the above-mentioned system stability analysis results.

[0236] 3.2) Tracking control under unified boundary constraints

[0237] To verify the effectiveness of the control algorithm in step 2.2), the MIMO non-linear system in step 3.1) is used as the simulation object. The cases (ii)-(iv) in step 2.1) are simulated and verified.

[0238] Simulation 3 (asymmetric semi-global performance constraints (ii) and (iii)): The desired trajectory is y d = [0.2cos(0.2πt), 0.2sin(0.2πt)] T . The initial value is set as x1(0) = [-1.3, 1.5]T where \(x_2(0)=[0,0]\) T and If the initial tracking error \(e\) j (0) directions are known (\(\text{sign}(e_1(0)) = -1\), \(\text{sign}(e_2(0)) = 1\)), the key design parameters can be selected as p \(\gamma_1 = 1\) and the other parameters for the unified preset performance boundary are selected as \(\rho\) 1,T \(=\rho\) 2,T \(= 0.2\), σ \(\gamma_1 = 0.35\), σ \(\gamma_2 = 0.45\), \(b_1 = b_2 = b_3 = 3\) and \(T = 2\). The parameters of the controller are selected as \(c_2 = 30\), \(\lambda_3 = 1\), \(\lambda_4 = 0.0001\), \(\beta = 0.01\) and \(\tau_1 = \tau_2 = 0.5\). The simulation results are as Figures 9 - 12 shown.

[0239] From Figure 10 it can be seen that only when the directions of \(e_1(0)\) and \(e_2(0)\) are known, the tracking error can converge to the preset accuracy within the preset time \(T = 2s\), and the overshoot is small enough. Figure 9 is the running curve of the desired trajectory and the actual output, Figure 11 is the running curve of the control input, Figure 12 is the running curve of the virtual parameter. It can be seen from this that all these signals are bounded, which is consistent with the previous system stability analysis results.

[0240] Simulation 4 (symmetric global performance constraint (iv)): In this simulation, the initial state of the system is set as \(x_1(0)=[5.2,5]\) T where \(x_2(0)=[0,0]\) T and If the initial directions of the tracking error \(e_1(0)\) and \(e_2(0)\) are unknown, the key design parameters can be selected as and The settings of the desired trajectory and other design parameters for the unified preset performance boundary are the same as those in Simulation 3. The controller parameters are selected as \(c_2 = 0.2\), \(\lambda_1 = 0.0001\), \(\lambda_2 = 1\), \(\beta_1 = 0.01\) and \(\tau_1 = \tau_2 = 0.5\). The simulation results are as Figures 13 - 16 shown.

[0241] From Figure 14It can be seen that the tracking errors e1 and e2 have been varying within the unified performance boundary. Since the performance boundary tends to infinity at the initial moment, a fast and accurate global performance tracking control can be achieved, and the control algorithm does not depend on the initial values of the system. Therefore, the parameter selection is more flexible and the control is easier to implement. Figure 13 is the running curve graph of the desired trajectory and the actual output, Figure 14 is the running curve graph of the control input, Figure 16 is the running curve graph of the virtual parameter. It can be seen from it that all these signals are bounded, which is consistent with the foregoing system stability analysis results.

[0242] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the technical solutions of the present invention, and they should all be covered within the scope of the claims of the present invention.

Claims

1. A design method for a high-precision preset performance controller under unified boundary constraints, characterized in that, The method comprises the following steps: 1) By introducing appropriate adjustment functions, precise preset performance boundaries are designed that can automatically match the initial constraints; 2) For different initial error values, a high-precision preset performance control algorithm under precise boundary constraints is proposed; 3) By establishing a suitable scale transformation function, a unified preset performance boundary is designed that can present various constraint forms under different parameter selections; 4) For different initial conditions, a high-precision preset performance control algorithm under unified boundary constraints is proposed.

2. The design method of a high-precision preset performance controller under unified boundary constraints according to claim 1, characterized in that In step 1), the accuracy constraint of the classical performance boundary is improved by introducing a suitable adjustment function, and an accurate preset performance boundary that can automatically match the initial constraint conditions is designed; the specific steps are as follows: Consider a class of uncertain MIMO nonlinear standard form systems: wherein, and are measurable system states, is an unknown time-varying nonlinear function, is a time-varying unknown asymmetric control gain matrix, is an unknown parameter vector, and are the control input and system output respectively, is an unknown bounded external disturbance; in addition, define as the tracking error, where is the desired trajectory; To make the tracking error enter the preset steady-state accuracy within the preset time, the finite-time performance function ρ j (t) is adopted. Such functions have the following properties: 1) ρ j (0) = 1 and ρ j (t) = ρ j,T for t ≥ T, where T > 0 is the convergence time that can be arbitrarily specified in advance by the user; 2) 3) is continuous for t ≥ 0; The finite-time performance function satisfying the above properties is designed as follows: where, ρ j,T and b1 are design parameters, ρ j,T > 0 is the maximum allowable steady-state error, b1 ≥ 1 is used to adjust the convergence rate of the function ρ j (t); To improve the accuracy constraint of the finite-time performance function (2), a suitable adjustment function is introduced This type of function has the following properties: 1) It is a strictly decreasing function on [0, t); 2) And when t ≥ T, there is T > 0 is the user-defined convergence time; 3) When t ≥ 0, there is It always holds; The adjustment function satisfying the above properties is designed as follows: where b2 is a design parameter, b2≥1 is used to adjust the convergence rate of the function ; it is not difficult to see that for 0≤t<T, the common form of the function also has etc.; According to (2) and (3), the design precisely presets the performance boundary as follows: Among them, and σ j are design parameters, e j (0) is the initial value of the tracking error for a single dimension, and are used to automatically match different initial error values, and σ j > 0 are the upper and lower bound constants related to the tracking accuracy, respectively; When the precise preset performance boundary is designed, it is used to constrain the tracking error e j (t) as follows: For different initial error values e j (0), if e j (t) satisfies the performance constraint inequality (5), then the following properties hold: 1) e j (t) enters the preset accuracy within the preset time; 2) the overshoot of e j (t) during the tracking process can be small enough; 3) the initial performance constraint conditions can be automatically matched.

3. The design method of a high-precision preset performance controller under unified boundary constraints according to claim 1, characterized in that, In step 2), a suitable augmented Lyapunov function is constructed with the help of an unknown but positive definite diagonal auxiliary matrix and a known performance matrix, and the filter variable method and the core function method are used to complete the design of a high-precision preset performance control algorithm under precise boundary constraints. The stability analysis and proof are performed by the Lyapunov direct method. The specific steps are as follows: In order to convert the performance constraint problem (5) into an unconstrained bounded problem, the nonlinear conversion function is adopted as follows: Among them, Introduce the filtering variable s j As follows: where β j > 0 is a design parameter; Therefore, the system model (1) can be transformed into: wherein, R is an additional matrix brought by considering the preset performance, is a known function; The robust adaptive controller is designed as follows: where \(c1 > 0\), \(\lambda1 > 0\) and \(\lambda2 > 0\) are design parameters, \(\psi1\) is a core function related to modeling uncertainties, is the estimated value of the unknown virtual parameter \(a1\), is arbitrarily selectable initial value of; According to the Lyapunov stability theory, the following conclusions can be drawn: for nonlinear systems (1), under reasonable assumptions, according to the robust adaptive controller (9), there are suitable design parameters, so that: for the time-varying asymmetric control gain matrix, under the condition of considering performance constraints, all signals in the closed-loop system (1) are bounded; the tracking error converges to a given accuracy within a preset time with a sufficiently small overshoot, and even if the expected trajectory changes or the system is restarted, the initial performance constraints can be automatically matched.

4. The design method of a high-precision preset performance controller under unified boundary constraints according to claim 1, characterized in that In step 3), in order to improve the versatility of the precise preset performance control algorithm in different scenarios, by establishing a suitable scale transformation function, the precise preset performance boundary is expanded to a unified preset performance boundary, so that a variety of performance constraints can be realized under a fixed control structure, and the tracking error can also be converged to the predetermined accuracy within a pre-set time under different initial conditions; the specific steps are as follows: First, construct the scale transformation function as follows: where \(q\) j \(\in \{0, 1\}\) is a design parameter, \(b_3\geq1\) is used to determine the convergence rate of the function \(\kappa(t)\), \(\kappa(t)\) is a strictly increasing function and satisfies \(\kappa(0) = 0\) and \(\kappa(T)=1\), \(T > 0\) is a user-defined convergence time, and \(\kappa(t)=\sin[(\pi / 2)(t / T)]\) is selected; other common forms of \(\kappa(t)\) are \(1-(1 - t / T)\) 2 , \(\cos[(\pi / 2)(t / T - 1)]\), etc.; it is not difficult to see that the scale transformation function \(\eta\) j (t) has the following properties: 1) when \(q\) j = 1, \(\eta\) j (t) satisfies \(\eta\) j (t)\(\equiv1\) on \([0, +\infty)\); 2) when \(q\) j = 0, \(\eta\) j (t) is monotonically increasing on \([0, T)\), satisfies \(\eta\) j (0) = 0 and \(\eta\) j (t) = 1 for \(t\geq T\); 3) regardless of the value of \(q\) j , \(\eta\) j (t) is continuously differentiable and bounded, and its first derivative is also bounded; In order to decouple the constraint bound from the tracking error, the transfer error is introduced as follows: χ j (t) = η j (t)e j (t) (11) Using the defined transfer error χ j (t), the precisely preset performance boundary designed in step 1) becomes: In order to achieve multiple performance constraints under a fixed control structure, a unified preset performance boundary is designed as follows: Among them, η j (t) and is the scaling function defined by (10), p j , q j and are design parameters; It should be noted that a very small positive constant can be embedded into the scaling function so that when and there is Otherwise, the boundary is infinite, thus leading to a singular problem at the initial moment; When the design parameter q j , p j , q j and take different values, there are the following five cases for the uniformly preset performance boundaries and : ① always positive and sometimes positive and sometimes negative; ② constantly negative and negative and positive at times; ③ t≥0； ④ t≥0； ⑤ t≥0； After designing a unified preset performance boundary, it is used to constrain the tracking error as follows: For different initial conditions (e j (0) being exactly known, direction known, and direction unknown), if the tracking error e j (t) satisfies the performance constraint inequality (14), then the following properties hold: 1) The performance constraints of e j (t) exhibit various forms under different parameter selections; 2) e j (t) converges to the predetermined accuracy within a preset time.

5. The high-precision preset performance controller design method under unified boundary constraints according to claim 1, characterized in that, In step 4), by reasonably using design parameters, the nonlinear transformation function for dealing with complex constraint conditions is improved, which not only reduces the number of boundary parameters that play an important role in the equivalent transformation, but also establishes an asymptotic equivalent relationship between the transformation variable and the constraint variable; with the help of an unknown but positive definite diagonal auxiliary matrix, a known performance matrix and its minimum eigenvalue, a suitable augmented Lyapunov function is constructed, and the high-precision preset performance control algorithm design under unified boundary constraints is completed using the filtered variable method and the core function method; the specific steps are as follows: The unified preset performance boundary (13) will present various constraint forms under different design parameters. In order to equivalently transform the performance constraint problem (14) into an unconstrained bounded problem, a special nonlinear transformation function is designed as follows: Among them, τ j > 0 is a design parameter; Introduce the filtered variable ξ as follows: Among them, β1, β2, …, β n-1 are design parameters, and their selection needs to satisfy a n-1 + β n-1 a n-2 + … + β1 is a Hurwitz polynomial; Thus, the system model (1) can be transformed into: where Γ is an additional matrix brought about by considering the preset performance, is a known function; Design the robust adaptive controller as follows: where \(c_2 > 0\), \(\lambda_3 > 0\), \(\lambda_4 > 0\) and \(\theta > 0\) are design parameters, \(\psi_2\) is a core function related to modeling uncertainties, is the estimated value of the unknown virtual parameter \(a_2\), is arbitrarily selected as the initial value of; According to the Lyapunov stability theory, the following conclusions can be obtained: for the nonlinear system (1), under reasonable assumption conditions, according to the robust adaptive controller (18), there exist suitable design parameters such that: for the time-varying asymmetric control gain matrix, considering the performance constraints, all signals in the closed-loop system (1) are bounded; multiple performance constraints can be achieved under a fixed control structure, and the tracking error can converge to the predetermined accuracy within the preset time under different initial conditions.

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