Prescribed Finite Time Stabilization of Linear Systems With State Constraints

This paper studies the prescribed finite time stabilization control problem of multi-input linear control systems subject to uncertainties and state constraints. Different from finite-time and fixed-time control methods, a novel switching nonsingular coordinate transformation function based on the block decomposition technique is developed, resulting in a solution that the control action is continuous and bounded based on the transformed system. Moreover, the boundedness of closed-loop system states is ensured with prescribed finite time performance. Furthermore, we incorporate the barrier Lyapunov function into the stabilization control design to restrain the amplitudes of system states. Numerical examples are provided to verify the effectiveness of the proposed prescribed finite time control strategy.


I. INTRODUCTION
Stabilization control is of special interest to linear control systems from the practical and theoretical perspectives. Effective methods have been proposed such as PID control, sliding mode control, model predictive control, backstepping, H ∞ control [1], [2], etc. In particular, sliding mode control is popular due to its effectiveness, the high-order sliding mode control algorithms with finite-time and fixed-time regulations have been applied in the fields of mechanical industries and electronics [3]- [5].
The finite-time and fixed-time stability is widely considered since the attribute of owing a specified time. The finite-time stability for second-order and high-order systems are respectively considered in [6] and [7]. The finite-time stabilization and optimal feedback control for the nonlinear dynamical system are investigated in [8]. The further endeavor made by [10] attains the finite-time stabilization of stochastic high-order nonlinear systems. The literature [11] proposes a constraint stabilization control law with higher-order sliding mode finite-time design, leading that the output of the system remains in some prescribed range. For the above literature, the settling time of finite-time control depends on both the initial conditions and the preselected control parameters. Polyakov [3] studied the nonlinear feedback controller for fixed-time stabilization of linear control The associate editor coordinating the review of this manuscript and approving it for publication was Bohui Wang . systems. In [13], the fixed-time backstepping control design for high-order strict-feedback nonlinear systems via terminal sliding mode is investigated. However, although the fixed-time control method removes the constraint of initial conditions, it can only converge the states before the given settling time, and the predescribed converge time cannot be guaranteed exactly. Song et al. in [1] and [12] consider the time-varying feedback stabilization of normal form nonlinear systems in prescribed finite time by introducing the time-scaling function. In [15], the prescribed-time observers for linear systems in observer canonical form are considered. Different from finite-time and fixed-time control methods, fractional-power terms are not utilized and system states converge to zero at the exact time T . However, the effective time only works for the time t ∈ [t 0 , t 0 + T ) and the design parameters should be chosen so that the matrix obtained from the characteristic polynomial is Hurwitz [1]. Moreover, the dynamics considered above are limited to the norm form or the observer canonical form. According to the block control principle in [3], for high-order subsystems transformed by a linear system with multiple inputs, the above algorithms are not applicable.
To overcome the limited time interval of t ∈ [t 0 , t 0 + T ), in [2] the prescribed-time consensus and containment control strategies of single integrator multi-agent systems not affected by disturbances are considered. In [14] by introducing a time-varying piecewise function the adaptive fault-tolerant prescribed-time control for nonlinear teleoperation systems with position error constraints is considered beyond the limited time interval. Authors in [18] considers the consensus control for wheeled mobile robots by designing the specified-time observer with time scale function under input saturation constraint. However, the continuity of time-varying scaling functions above is not guaranteed. Furthermore, due to the introduction of 1 T −t , the input boundedness and state constraint issues are necessary to be investigated.
Motivated by the discussions above, we address the nonlinear feedback design for prescribed finite time stabilization of multi-input linear control systems with state constraints. Firstly, in view of the difficulties caused by the distinct subsystem dimensions of multi-input linear systems, a novel switching nonsingular coordinate transformation function based on the block decomposition technique is proposed. Compared with finite time methods [6]- [11] and the fixed time method [3], due to the proposed high-power time scaling function, the convergence time T of closed-loop system states can be explicitly predetermined without the limitation of system initial values. Secondly, the operation time of the control scheme is extended to t ∈ [0, ∞) compared with the previous works [1], [12] and [16], and the continuity of the switching time-varying scaling function is guaranteed. Moreover, the input boundedness is proven in spite of the existence of term 1 T −t . Thirdly, the state constraint issue is considered by the barrier Lyapunov function, and the amplitudes of system states can be restrained.
The organization of this article is organized as follows. Section 2 presents some preliminaries and problem statements. Section 3 elaborates the prescribed time stability control design of the linear systems. Section 4 gives the numerical simulation to verify the theoretical analysis. Finally, Section 5 concludes the paper.
Notations : Through this article, x signifies its standard Euclidean norm.
x ∞ = ess sup t>p |x| and x p is the p norm. Let R be the set of the real number, R n be the set of the n-dimensional vector and R n×n be the set of the n-dimensional matrix. The superscript T means the transpose for real matrices and ''+'' means the pseudo-inverse for real matrices. rown(W ) is the row number of the matrix W , ker(W ) and range(W ) are the null space and the column space of W , and null(W ) is the matrix with columns defining the orthonormal basis of ker(W ).

II. PROBLEM FORMULATION
Consider the following class of uncertain multi-input linear systemsẋ where x ∈ R n is the system state, u ∈ R m is the control input, matrices A ∈ R n×n and B ∈ R n×m are the system matrix and control gain matrix. The term τ d (t, x(t)) = Bχ(t, x(t)) represents the matched uncertainties. Assumption 1: The matrix pair (A, B) is controllable.

Assumption 2:
The term χ (t, x(t)) ≤ χ 0 (t, x(t)) for ∀t > 0 and ∀x ∈ R n , where χ 0 (t, x(t)) is a known function. Remark 1: The observer of the uncertainty with an unknown upper bound has been discussed to improve the system transient performance in our previous work [16]. In fact, the observer design has been discussed warmly, the boundary of the uncertainty is generally unknown, and the norm value of its derivative is assumed to have an unknown upper bound. For instance, in [31] the fixed-time observer is designed to compensate for such uncertainties. Before moving on, we give the following lemma which will be used in the proof of the main results in the paper.
Lemma 1 [3]: If the pair (A, B) is controllable, then based on the algorithm below, Then orthogonal matrix G can be given as with w i = n − rown(T i ). According to the coordinate transformation y = Gx, system (1) can be reduced to the block form         ẏ 1 = A 1,1 y 1 + A 1,2 y 2 y 2 = A 2,1 y 1 + A 2,2 y 2 + A 2,3 y 3 · · · y k = A k,1 y 1 + · · · + A k,k y k + A k,k+1 (u + χ) where A i,j are derived from the matrix GAG T . Lemma 2 [3]: Consider the dynamics in (1) and (3), combining with the coordinate transformation s = y and the inverse transformation y = −1 s below A rj y j ) (4) Then system (1) can be redescripted as the following block subsystem form where the matrices A i,j andÃ i,j can be derived from the matrix G. Remark 2: According to the algorithm in Lemma 1, it is noted that the dimensions of subsystems y 1 , . . . , y k in (3) and s 1 , . . . , s k in (6) may be different. Due to the input matrix A k,k+1 with the structure of full row rank, it brings difficulty to design the stabilization controller of multi-input linear systems.
Objective: This paper aims to solve the prescribed finite time stabilization control for multi-input linear systems (1) with both uncertainties and state constraints. We will employ prescribed finite time state feedback controller for systems (1) by a novel switching nonsingular coordinate transformation function based on block decomposition technique.
Remark 3: Compared with the finite time methods [6]- [11] and the fixed time method [3], the convergence time T can be conveniently set by introducing the term 1 T −t , which can be considered as the role of gain term acting on the control input and adjusting the convergence rate. Besides, different from [1] and [12], the stabilization system of (1) owns the structure of multiple inputs, and the controller proposed in this paper can be obtained without solving certain polynomials like [12]. Moreover, the case of time interval t > T and the control continuity are not investigated in [1], [12] and [16]. Due to the existence of the term 1 T −t introduced by the time scale function, it is necessary to analyze the input boundedness and state constraint issues.

III. MAIN RESULTS
This section presents the design and analysis of the prescribed finite time stabilization controller for the multi-input linear system. First, the novel switching coordinate transformation and inverse coordinate transformation are proposed. Second, stabilization control laws are developed and the proof of boundedness is given. Finally, the stabilization control law with state constraints is analyzed via the barrier Lyapunov function.

A. COORDINATE TRANSFORMATION AND INVERSE COORDINATE TRANSFORMATION
To obtain the prescribed-time controller, the switching scaling function is introduced as follows where the parameter T is the prescribed finite time satisfying 0 < T p ≤ T (T p is the physically possible time interval which represents the time consuming of signal transmission and processor computing) and is irrelevant to system initial conditions. The parameter δ is a positive value satisfying that 0 < δ 1. It is noted that the function µ 1 (t, T ) meets that µ 1 (0, T ) = 1 (T +δ) γ , and when t = T , µ 1 (t, T ) = 1/δ γ with an integer γ greater than 1, which guarantees the continuity of the switching function (7). The function µ 1 (t, T ) is a positive non-decreasing function and is continuous everywhere. Moreover, when δ = 0 it becomes the case studied in [12].
Following the above function, we propose a coordinate transformation based on dynamics (6).
Consider the block subsystem (6), design the coordinate transformation w = Ps as follows, where the coefficients a i,j is a constant designed as Remark 4: Compared with [1], [12] and [16], the case of time interval t > T is considered. In terms of equations (8) and (9), the parameter δ is introduced to guarantee the continuity of state w i for t ∈ [0, ∞).
Theorem 1: For the coordinate transformation (8-11), the dynamics of w i , 1 ≤ i ≤ k, can be described aṡ where and the parameter k 1 is a positive value. Proof: The proofs are divided into two parts associated with above two cases in (8) and (9).
Step 1: The case of 0 ≤ t < T is firstly considered. According to equation (12), the desired transformed system is given below, Then w i+1 can be obtained as follows, (T −t+δ) γ is chosen carefully to guarantee the solvability and uniqueness of control input u. With the switching coordinate transformation (8), the derivative of w i along (14) is obtained aṡ (16) Then one has the solution of (15) as With the condition (10), we have Rewrite the above equation as follows, According to the equation (8), one gets the condition (11).
Step 2: The case of t ≥ T is considered. Then w i+1 can be obtained as follows, With the dynamics (12), the desired transformed system iṡ By the similar steps above, we have According to the equation (9), the condition (11) is obtained. The proof is completed. Remark 5: Different from [16], this paper designs the time-based power function (8) and (9). The continuity of the switching function of (7) is guaranteed and it owns the ability to adjust the transient response performance.
Lemma 3: For the coordinate transformation (8)(9)(10)(11), the inverse coordinate transformation s = P −1 w can be described as follows, where the coefficient b i,j is a constant designed as The proofs are divided into two parts associated with the above two cases in (21) and (22).
Step 1: The case of 0 ≤ t < T is firstly considered. The derivative of dynamic (21) with b i,q = 0 and b i,0 = 0 is given as then the solution of s i+1 gives By the coordinate transformation (21), it gives Then the equation (24) is obtained.
Step 2: The case of t ≥ T is considered. Similar to the above equations, one gets By the coordinate transformation (22), the condition (24) is obtained. The proof is completed.
Remark 6: According to the block decomposition technique in Lemma 1 and Lemma 2, the original system (1) is transformed into the system (6), which owns the structure of different dimensions of subsystems. The coordinate transformation proposed in (8)(9)(10)(11) is utilized to obtain the transformed system (12) from (6). With the transformed system (12), if the state w i , i = 1, . . . , k − 1 is bounded, then the state w i−1 can be bounded in prescribed time T . With the nonsingular coordinate transformations introduced in Lemma 1 and 2, the state x of the system (1) will be bounded in prescribed time T .
Remark 7: As we all know, there are countless solutions of w i+1 and s i+1 due to are selected specially for coordinate transformation and controller design in the next subsection.

B. PRESCRIBED-TIME STABILIZATION WITHOUT STATE CONSTRAINTS
Recalling the above coordinate transformations, we give the control strategy as below Then the block subsystems w can be given as Theorem 2: Consider the dynamics (1) with transformations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) and control law (25), the transformed system states w and s, the original system state x and control input u are prescribed finite time bounded.
Proof: Applying the coordinate transformations (8) and (9), we have (27) Then the derivative of w k along systems (6) is obtained aṡ By combining the controller (25) with the above equation, the dynamics (26) is obtained. Next, the stabilization property of system (26) is analyzed. Denote the candidate Lyapunov function V i = w T i w i 2 , i = 1, . . . , k.
Step 1: The case of i = k. The derivative of V k along (26) iṡ V k = − k 1 (φ(t)) γ w T k w k + 1 (φ(t)) γ w T k A k,k+1 χ (t, w). By applying Young's inequality with constant parameter 0 < λ < k 1 , we havė When t < T , we have It is known that the first term above is monotonically decreas- , and one has As t goes to T , we have lim The parameter δ is chosen as 0 < δ 1, we know that if δ is close to zero, lim t→T φ γ −1 = +∞ and lim Subsequently, the state w k is bounded as Step 2: The case of i = k − 1. Due to the fact thaṫ Then we have When t < T , we have Recalling equation (34), we have then it leads to Then one holds Since state w k and parameters T , k 1 are bounded, ϒ k−1 is bounded. Similarly, we have By the same steps, we have with ϒ k−1 = (1+ A k,k+1 exp( −k 1 δ + 1)). By the same steps, we have w k−q , q = 3, . . . , k − 1 are bounded. With the inverse coordinate transformation (21) and (22), then and the control input (25) is rewritten as According to the coordinate transformations (8) and (9), one has w k is bounded, then − s k φ , k−1 j=1 a k,j (φ) k+γ −j s j and Then the control input is bounded. The proof is completed.
Remark 8: The boundary value k,1 of the state w k in (30) is related to the magnitude of the uncertainties χ 0 , and it can be adjusted by choosing the parameters k 1 . Indeed, the system uncertainty has a significant influence on the control performance. The advanced nonlinear control methods in [24] and [25] can provide a good solution for dealing with various uncertainties.

C. PRESCRIBED-TIME STABILIZATION WITH STATE CONSTRAINTS
In this section, the barrier Lyapunov function is proposed for restraining the amplitudes of system states. Before moving on, we give the following lemma.
Lemma 4: [22] For any positive constant k w , the following inequality holds for any vector x ∈ R n in the interval x < k w , where log(•) denotes the natural logarithm of •.
Proof: Define the BLF Lyapunov function candidate as Hence, the states w k in (44) with the boundary w k < k wk is guaranteed if V k is made bounded. Due to equations (38) and (39), states w q , q = 1, . . . , k − 1 are also bounded. The derivative of V k along (44) is given aṡ With Lemma 4, we have log Then according to controller (45) and Young's inequality, the derivative of V k can be rewritten aṡ > 0. Similar to (28), we have that the variable w k is bounded and remains in the sets w i < k wk , if the initial value of w i (0) is in the set w i (0) < k wk . Then recalling (44), states w i , i = 1, . . . , k − 1 are also bounded. With s = Gx, s = P −1 w, and G are nonsingular transformation, we can know that the states of x and s are prescribed-time bounded and limited by x ∞ < P −1 G 2 2 max(k wi ). The proof is completed.
Remark 9: With the proposed controller (45) the state w k is designed to be bounded in prescribed time by (44), and then states w i , i = 1, . . . , k − 1 are bounded in prescribed time. In fact, the barrier Lyapunov function considered in (46) can only ensure the constant constraint, the time-varying barrier Lyapunov function which is close to practical applications can be referred to [23]. Moreover, the high-order full-state constraints can be solved based on [19]- [21], which will be our future work.

IV. SIMULATION
In this section, we verify the effectiveness of the proposed stabilization control methods in (25) To get fair comparison with [16], we use the same initial values as x 0 = (0.5, 0.2, 0.3) T . The cases γ = 1 to γ = 3 are chosen in (8) and (9), respectively. Compared with case γ = 1 in [16], Figures 1-3 show the evolutions of states for system (1). It is clearly shown that states x i , i = 1, 2, 3 can converge to the neighbor of the origin in the desired prescribed time T = 1s, which proofs the proposed   controller (25) is effective when 0 ≤ t < T and t ≥ T . Moreover, the transient response performance can be effectively improved by choosing a larger γ , which confirm that the high power nonsingular coordinate transformations (8)- (9) and (21)-(22) are significant and necessary. Figure 4 presents the plots of controller (25) and the boundedness of the controller is proved. In Figures 5-7, it can been seen that the amplitude of state x can be limited. To further show the effectiveness of the controller (45), we assume the constraint bound of w k as k wk = 0.18. In Figures 8-9, the value γ   is given as γ = 2. In Figure 8, we assume the system uncertainty τ d = 0, and the comparison of state response w 2 2 with and without the constraint bound of k wk = 0.18 is shown. With equation (49), the dimension of state w 2 is 2. From Figure 8, it illustrates that the state w 2 2 will converge to zero without the existence of the uncertainty τ d . Moreover, the controller proposed in (45) can bring the state w 2 2 into the specified constraint range k wk = 0.18, while the state w 2 2 with controller (42) will be out of limitation range. In Figure 9, we assume the uncertainty τ d as τ d = [2; 0; −5] * sin(t), and the comparison of state response w 2 2 with and without the constraint bound of k wk = 0.18 is shown. From Figure 9, it shows that the state w 2 2 can converge into the specified constraint range k wk = 0.18 with the controller (45). It is proven that the proposed controllers are effective.

V. CONCLUSION AND FUTURE WORK
In this paper, based on the block decomposition technique and novel switching nonsingular coordinate transformation methods, the prescribed finite time stabilization control laws for multi-input linear control systems are presented, which fills the gap of the control time interval in the case of t > T while guaranteeing continuity and boundedness of the controller. Furthermore, the amplitudes of system states are restrained with constant boundary value by introducing the barrier Lyapunov function. Future research work will focus on the extension of the results to the prescribed-time consensus problem of multiple linear systems with time-varying communication delay inspired by [27]- [30]. Furthermore, the event trigger prescribed time control of linear systems which can reduce the burden of actuators will be another future work.