On Switched Control Design of Linear Time-Invariant Systems with Polytopic Uncertainties

Abstract

This paper proposes a new switched control design method for some classes of linear time-invariant systems with polytopic uncertainties. This method uses a quadratic Lyapunov function to design the feedback controller gains based on linear matrix inequalities (LMIs). The controller gain is chosen by a switching law that returns the smallest value of the time derivative of the Lyapunov function. The proposed methodology offers less conservative alternative than the well-known controller for uncertain systems with only one state feedback gain. The control design of a magnetic levitator illustrates the procedure.

1. Introduction

In recent years, there has been much interest in studying switched systems, due to the considerable advance in this research field, initiating mainly with [1–4]. For linear time-invariant systems, the transient response can be improved through switching controllers [5], as can be seen, for instance, in [6–9].

In general, most papers in the area of switched linear systems utilize multiple Lyapunov functions [10–14]. A design method that is applicable to a large class of switched controllers for linear systems with input signals, formulated with bilinear matrix inequalities (BMIs), is proposed in [10]. The switching law defines regions where different subsystems are activated, resulting in a switched linear system that is exponentially stable. Study results on the stability analysis and stabilization of switched systems can be seen in [11], which presents necessary and sufficient conditions for asymptotic stability. Moreover, the problem of switching stabilizability is studied, investigating under what conditions it is possible to stabilize a switched system by designing switching control laws. Necessary and sufficient conditions for switched linear systems with polytopic uncertainties to be quadratically stabilizable via state feedback can be found in [13].

The design of the robust state feedback control for continuous-time systems subject to norm bounded uncertainty can be seen in [12], where the switching rule, as well as the state feedback gains, is determined from the minimization of a guaranteed cost function derived from a multiobjective criterion. The paper [14] presents a generalization of the results proposed in [12] and offers a procedure that finds, simultaneously, a set of state feedback gains and a switching rule to orchestrate them, rendering the equilibrium point of closed-loop system globally asymptotically stable for all time-varying uncertain parameters under consideration and assuring a guaranteed ℋ₂ cost.

Although outnumbered, there are papers about switched linear systems using a common Lyapunov function as in [15, 16]. In [15], the stability of switched linear systems with polytopic uncertainties was studied. Some criteria for globally exponential stability were also established, in which all the vertex matrices of the switched systems are commutative pairwise, thus generalizing the existing results for switched linear systems without uncertainty. In [16] the quadratic stability for continuous-time and discrete-time switched linear systems with a switching rule using a single symmetric positive definite matrix, which depends on system states vector, was studied.

This paper proposes a new methodology for switched control design of a class of linear systems with polytopic uncertainties. This method uses a common Lyapunov function and quadratic stability for designing the state feedback controller gains based on linear matrix inequalities (LMIs). The proposed controller chooses a gain from a set of gains by means of a suitable switching law that returns the smallest value of the Lyapunov function time derivative. The proposed methodology allows a less conservative LMI-based design than the traditional method for uncertain plants that considers only one state feedback controller gain [17].

To confirm the advantages of the proposed methodology, figures comparing the regions of feasibility of the proposed methodology with the one state feedback gain classical method are presented. To compare the performance of the proposed control law with the classical control law, an application in the magnetic levitator control was simulated. The computational implementations were carried out using the modelling language YALMIP [18] with the solver SeDuMi [19].

For convenience, in some places, the following notation is used: 𝕂_𝑟={}(𝑡)𝑉(𝑥(𝑡‖𝑥‖1,2,…,𝑟,𝑥=𝑥,))=𝑉,₂=𝑥^𝑇(𝛼)=𝑥,𝐴,𝐵,𝐾)(^𝑟_𝑖=1𝛼_𝑖𝐴_𝑖,𝐵_𝑖,𝐾_𝑖,with𝛼_𝑖≥0,_𝑟𝑖=1𝛼_𝑖=1,𝛼=𝛼₁,𝛼₂,…,𝛼_𝑟^𝑇, (1)(2) where 𝑟=2^𝑠 and 𝑠 is the number of uncertain parameters in the plant.

2. Linear Systems with Polytopic Uncertainties

Consider the linear system with polytopic uncertainties 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵(𝛼)𝑢(𝑡), (3) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝐴(𝛼) and 𝐵(𝛼) as in (2), with 𝐴_𝑖∈ℝ^𝑛×𝑛 and 𝐵_𝑖∈ℝ^𝑛×𝑚 for 𝑖∈𝕂_𝑟.

Assuming that all state variables are available for feedback, the control law largely used in the literature is given by [17]: 𝑢(𝑡)=−𝐾𝑥(𝑡), (4) where 𝐾∈ℝ^𝑚×𝑛. Replacing (4) in (3), one obtains the feedback system 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)−𝐵(𝛼)𝐾𝑥(𝑡)=^𝑟_𝑖=1𝛼_𝑖𝐴_𝑖−𝐵_𝑖𝐾𝑥(𝑡). (5)

Define a feedback control law with the state vector as 𝑢(𝑡)=𝑢_𝛼(𝑡)=−^𝑟_𝑖=1𝛼_𝑖𝐾_𝑖𝑥(𝑡)=−𝐾(𝛼)𝑥(𝑡), (6) where 𝐾_𝑖∈ℝ^𝑚×𝑛, 𝑖∈𝕂_𝑟. Considering (2) and from (6) and (3), 𝑥̇(𝑡)(𝛼)𝑥(𝑡)(𝛼)𝐾(𝛼)𝑥(𝑡)==𝐴−𝐵_{𝑟𝑟𝑖=1𝑗=1}𝛼_𝑖𝛼_𝑗𝐴_𝑖−𝐵_𝑖𝐾_𝑗𝑥(𝑡). (7)

2.1. Stability of Linear Systems via LMIs

This section presents some results on stability and control of linear systems with polytopic uncertainties.

Theorem 1 (see [20]). The linear system with polytopic uncertainties given in (5) is quadratically stabilizable if and only if there exist a common symmetric positive definite matrix 𝑋 and 𝑀∈ℝ^𝑚×𝑛 such that, for all 𝑖∈𝕂_𝑟, 𝑋𝐴^𝑇_𝑖+𝐴_𝑖𝑋−𝐵_𝑖𝑀−𝑀^𝑇𝐵^𝑇_𝑖≺0. (8) If there exists such a solution, the controller gain is given by 𝐾=𝑀𝑋⁻¹.

Theorem 2. The equilibrium point 𝑥=0 of the linear system with polytopic uncertainties given in (7) is asymptotically stable in the large if there exist a common symmetric positive definite matrix 𝑋 and 𝑀_𝑖∈ℝ^𝑚×𝑛 such that, for all 𝑖,𝑗∈𝕂_𝑟, the following LMIs hold: 𝑋𝐴^𝑇_𝑖+𝐴_𝑖𝑋−𝐵_𝑖𝑀_𝑖−𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝐴≺0,_𝑖+𝐴_𝑗𝑋+𝑋𝐴_𝑖+𝐴_𝑗^𝑇−𝐵_𝑖𝑀_𝑗−𝐵_𝑗𝑀_𝑖−𝑀^𝑇_𝑖𝐵^𝑇_𝑗−𝑀^𝑇_𝑗𝐵^𝑇_𝑖⪯0,𝑖<𝑗. (9) If (9) are feasible, the controller gains are given by 𝐾_𝑖=𝑀_𝑖𝑋⁻¹, 𝑖∈𝕂_𝑟.

Proof. Consider a quadratic Lyapunov candidate function 𝑉=𝑥^𝑇𝑃𝑥. Thus, from (7) note that 𝑉̇=𝑥̇^𝑇𝑃𝑥+𝑥^𝑇𝑃𝑥̇=^𝑟𝑟_𝑖=1𝑗=1𝛼_𝑖𝛼_𝑗𝑥^𝑇𝐴_𝑖−𝐵_𝑖𝐾_𝑗^𝑇𝑃𝑥+𝑥^𝑇𝑃^𝑟𝑟_𝑖=1𝑗=1𝛼_𝑖𝛼_𝑗𝐴_𝑖−𝐵_𝑖𝐾_𝑗𝑥=𝑥^𝑇𝑟_𝑖=1𝛼²_𝑖𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖−𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃−𝑃𝐵_𝑖𝐾_𝑖𝑥+𝑥^{𝑇𝑟−1𝑟}_{𝑖=1𝑗=1+𝑖}𝛼_𝑖𝛼_𝑗𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖+𝐴^𝑇_𝑗𝑃+𝑃𝐴_𝑗−𝐾^𝑇_𝑗𝐵^𝑇_𝑖𝑃−𝑃𝐵_𝑖𝐾_𝑗−𝐾^𝑇_𝑖𝐵^𝑇_𝑗𝑃−𝑃𝐵_𝑗𝐾_𝑖𝑥. (10) Now, 𝛼_𝑖≥0,𝑖∈𝕂_𝑟 and ∑^𝑟_𝑖=1𝛼_𝑖=1. Then, from (10), 𝑉̇<0 (for 𝑥≠0) if for 𝑖,𝑗∈𝕂_𝑟𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖−𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃−𝑃𝐵_𝑖𝐾_𝑖𝐴≺0,_𝑇𝑖𝑃+𝑃𝐴_𝑖+𝐴^𝑇_𝑗𝑃+𝑃𝐴_𝑗−𝐾^𝑇_𝑗𝐵^𝑇_𝑖𝑃−𝑃𝐵_𝑖𝐾_𝑗−𝐾^𝑇_𝑖𝐵^𝑇_𝑗𝑃−𝑃𝐵_𝑗𝐾_𝑖⪯0,𝑖<𝑗. (11)
Defining 𝑋=𝑃⁻¹, 𝑀_𝑖=𝐾_𝑖𝑋 and pre- and postmultiplying (11) by 𝑋, one obtains (9). The proof is concluded.

Corollary 3. If 𝐵₁=𝐵₂=⋯=𝐵_𝑟=𝐵, then the equilibrium point 𝑥=0 of the linear system with polytopic uncertainties given in (7) is asymptotically stable in the large if there exist a symmetric positive definite matrix 𝑋 and 𝑀_𝑖∈ℝ^𝑚×𝑛, such that for all 𝑖∈𝕂_𝑟, 𝑋𝐴^𝑇_𝑖+𝐴_𝑖𝑋−𝐵𝑀_𝑖−𝑀^𝑇_𝑖𝐵^𝑇≺0. (12) If (12) is feasible, the controller gains are given by 𝐾_𝑖=𝑀_𝑖𝑋⁻¹, 𝑖∈𝕂_𝑟.

Proof. It is similar to the proof of Theorem 2, considering 𝐵_𝑖=𝐵, 𝑖∈𝕂_𝑟, and noting that now (7) can be rewritten as 𝑥̇(𝑡)=∑^𝑟_𝑖=1𝛼_𝑖(𝐴_𝑖−𝐵𝐾_𝑖)𝑥(𝑡).

In a control design, it is important to assure stability and usually other indices of performance for the controlled system, such as the setting time, constraints on input control and output signals. The setting time is related to the decay rate of the system (5), (or the largest Lyapunov exponent) which is defined as the largest 𝛽>0 such that lim_𝑡→∞𝑒^𝛽𝑡‖𝑥(𝑡)‖₂=0 (13) holds for all trajectories 𝑥(𝑡). As in [17, page 66], one can use a quadratic Lyapunov function 𝑉=𝑥^𝑇𝑃𝑥 to establish a lower bound 𝛽 for the decay rate, considering the condition 𝑉̇≤−2𝛽𝑉 for all trajectories 𝑥.

Theorem 4 (see [17]). The linear system with polytopic uncertainties given in (5) is quadratically stabilizable, with decay rate greater than or equal to 𝛽, if and only if there exist a symmetric positive definite matrix 𝑋 and 𝑀∈ℝ^𝑚×𝑛 such that, for all 𝑖∈𝕂_𝑟, the following LMIs are satisfied: 𝑋𝐴^𝑇_𝑖+𝐴_𝑖𝑋−𝐵_𝑖𝑀−𝑀^𝑇𝐵^𝑇_𝑖+2𝛽𝑋≺0. (14) If there exists such a solution, the controller gain is given by 𝐾=𝑀𝑋⁻¹.

Proof. It is similar to that of Theorem 1, considering 𝑉̇≤−2𝛽𝑉 [17].

Theorem 5. The equilibrium point 𝑥=0 of the linear system with polytopic uncertainties given in (7) is asymptotically stable in the large, with decay rate greater than or equal to 𝛽, if there exist a common symmetric positive definite matrix 𝑋 and 𝑀_𝑖∈ℝ^𝑚×𝑛 such that, for all 𝑖,𝑗∈𝕂_𝑟, 𝑋𝐴^𝑇_𝑖+𝐴_𝑖𝑋−𝐵_𝑖𝑀_𝑖−𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝐴+2𝛽𝑋≺0,_𝑖+𝐴_𝑗𝑋+𝑋𝐴_𝑖+𝐴_𝑗^𝑇−𝐵_𝑖𝑀_𝑗−𝐵_𝑗𝑀_𝑖−𝑀^𝑇_𝑖𝐵^𝑇_𝑗−𝑀^𝑇_𝑗𝐵^𝑇_𝑖+4𝛽𝑋⪯0,𝑖<𝑗. (15) If (15) are feasible, the gains are given by 𝐾_𝑖=𝑀_𝑖𝑋⁻¹,𝑖∈𝕂_𝑟.

Proof. The proof is similar to that of Theorem 2, considering 𝑉̇≤−2𝛽𝑉.

One can constraint the norm of the controller gains by imposing restrictions on 𝑀_𝑖,𝑖∈𝕂_𝑟, and 𝑋⁻¹ as in [21]. Thus, given the constants 𝜂>0 and 𝜂_𝑥>0, imposing that 𝑀^𝑇_𝑖𝑀_𝑖≺𝜂𝐼_𝑛,𝑖∈𝕂_𝑟, and 𝑋⁻¹≺𝜂_𝑥𝐼_𝑛, then a constraint on the controller gains may be established by the following theorem [21].

Theorem 6 (see [21]). The constraint on the norm of the controller gains such that 𝐾_𝑖𝐾^𝑇_𝑖≤𝜂𝜂²_𝑥𝐼_𝑚,𝑖∈𝕂_𝑟 is enforced if there exist constants 𝜂>0 and 𝜂_𝑥>0, such that the LMIs from Theorems 2 or 5 (or 1 or 4 replacing 𝐾_𝑖=𝐾 and also 𝑀_𝑖=𝑀), with the LMIs below hold: 𝜂_𝑥𝐼_𝑛𝐼_𝑛𝐼_𝑛𝑋≽0,𝜂𝐼_𝑛𝑀^𝑇_𝑖𝑀_𝑖𝐼_𝑚≽0,𝑖∈𝕂_𝑟. (16)

Proof. The proof is similar to that presented in [21].

3. Case 1: Linear Systems with a Constant Matrix 𝐵(𝛼)=𝐵

In this section, the design of a switched controller for the uncertain system (3) is proposed, assuming that 𝐵(𝛼)=𝐵 is a constant matrix, now given by 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵𝑢(𝑡). (17)

Suppose that (12) is feasible for all 𝑖∈𝕂_𝑟, and let 𝐾_𝑖=𝑀_𝑖𝑋⁻¹,𝑖∈𝕂_𝑟, the gains of the controller given in (6), and 𝑃=𝑋⁻¹ obtained from the conditions of Corollary 3. Then, define the switched controller 𝑢(𝑡)=𝑢_𝜎(𝑡)=−𝐾_𝜎𝑥(𝑡),𝜎=argmin_{𝑖∈𝕂𝑟}−𝑥(𝑡)^𝑇𝑃𝐵𝐾_𝑖𝑥(𝑡),𝜎∈𝕂_𝑟. (18)

Note that, in (18) for a given 𝑡, the switching law 𝜎 can be obtained by calculating −𝑥(𝑡)^𝑇𝑃𝐵𝐾_𝑖𝑥(𝑡), for 𝑖∈𝕂_𝑟, and then considering the index set Ω(𝑡)={𝑗∈𝕂_𝑟∶−𝑥(𝑡)^𝑇𝑃𝐵𝐾_𝑗𝑥(𝑡)≤−𝑥(𝑡)^𝑇𝑃𝐵𝐾_𝑖𝑥(𝑡),∀𝑖∈𝕂_𝑟}. The switching index 𝜎 can be given by 𝜎(𝑡)=𝑗∈Ω such that 𝑗≤𝑖, for all 𝑖∈Ω(𝑡).

The implementation of (18) in a control application can be done using analog and/or digital electronics. Further details on this subject can be found, for instance, in [22, 23].

Therefore, from (2), the controlled system (17) and (18) is given by 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵𝑢_𝜎(𝑡)=^𝑟_𝑖=1𝛼_𝑖𝐴_𝑖−𝐵𝐾_𝜎𝑥(𝑡). (19)

Theorem 7. Assume that the conditions of Corollary 3, related to the system (17) with the control law (6), hold and obtain 𝐾_𝑖=𝑀_𝑖𝑋⁻¹, 𝑖∈𝕂_𝑟, and 𝑃=𝑋⁻¹. Then, the switched control law (18) makes the equilibrium point 𝑥=0, of the system (17), asymptotically stable in the large.

Proof. Consider a quadratic Lyapunov candidate function 𝑉=𝑥^𝑇𝑃𝑥. Define 𝑉̇_𝑢𝛼 and 𝑉̇_𝑢𝜎, the derivatives of 𝑉 for the system (17), with the control laws (6) and (18), respectively. Then, from (17) and (18), 𝑉̇_𝑢𝜎=2𝑥^𝑇𝑃𝑥̇=2𝑥^𝑇𝑃𝐴(𝛼)𝑥+𝐵𝑢_𝜎=2𝑥^𝑇𝑃𝐴(𝛼)𝑥+2𝑥^𝑇𝑃𝐵−𝐾_𝜎𝑥. (20) From (2), ∑^𝑟_𝑖=1𝛼_𝑖=1 and 𝛼_𝑖≥0,𝑖∈𝕂_𝑟. Thus, note that min_{𝑖∈𝕂𝑟}𝑥^𝑇𝑃𝐵−𝐾_𝑖𝑥≤𝑥^𝑇𝑃𝐵−^𝑟_𝑖=1𝛼_𝑖𝐾_𝑖𝑥, (21) and from (20), the switching law given in (18) and (6), observe that 𝑉̇_𝑢𝜎=2𝑥^𝑇𝑃𝐴(𝛼)𝑥+2min_{𝑖∈𝕂𝑟}𝑥^𝑇𝑃𝐵−𝐾_𝑖𝑥≤2𝑥^𝑇𝑃𝐴(𝛼)𝑥+2𝑥^𝑇𝑃𝐵−^𝑟_𝑖=1𝛼_𝑖𝐾_𝑖𝑥=2𝑥^𝑇𝑃(𝐴(𝛼)−𝐵𝐾(𝛼))𝑥=2𝑥^𝑇𝑃𝐴(𝛼)𝑥+𝐵𝑢_𝛼=𝑉̇_𝑢𝛼. (22) Therefore, 𝑉̇_𝑢𝜎(𝑥(𝑡))≤𝑉̇_𝑢𝛼(𝑥(𝑡)) and the proof is concluded.

Remark 8. Theorem 7 shows that if the conditions of Corollary 3 are satisfied, then 𝑉̇_𝑢𝛼(𝑥(𝑡))<0 for all 𝑥(𝑡)≠0 and therefore 𝑉̇_𝑢𝜎(𝑥(𝑡))<0 for 𝑥(𝑡)≠0, ensuring that the equilibrium point 𝑥=0 of the controlled system (17) and (18) is asymptotically stable in the large. Thus, Corollary 3 can be used to project the gains 𝐾₁,𝐾₂,…,𝐾_𝑟 and the matrix 𝑃=𝑋⁻¹ of the switched control law (18). Additionally, note that the switched control law (18) does not use the uncertain variables 𝛼_𝑖, 𝑖∈𝕂_𝑟, which would be necessary to implement the control law (6). Furthermore, it also offers an alternative less conservative than the well-known control law for uncertain systems presented in (4), with only one controller gain 𝐾.

4. Case 2: Linear System with an Uncertain Matrix 𝐵(𝛼)

In this case, the linear system with polytopic uncertainties will be considered as given in (3); with 𝛼_𝑖,𝑖∈𝕂_𝑟, defined in (2), namely, 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵(𝛼)𝑢(𝑡),𝐴(𝛼)=^𝑟_𝑖=1𝛼_𝑖𝐴_𝑖,𝐵(𝛼)=^𝑟_𝑖=1𝛼_𝑖𝐵_𝑖. (23)

Let 𝑣(𝑡)∈ℝ^𝑚 be the time derivative of the control input vector 𝑢(𝑡)∈ℝ^𝑚. Define 𝑥_𝑛+𝑙(𝑡) and 𝑣_𝑙(𝑡), such that 𝑥̇_𝑛+𝑙(𝑡)=𝑢̇_𝑙(𝑡)=𝑣_𝑙(𝑡),𝑙=1,2,…,𝑚. Thus, one obtains the following system: 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵(𝛼)𝑢(𝑡),𝑥̇_𝑛+1(𝑡)=𝑣₁(𝑡),⋮𝑥̇_𝑛+𝑚(𝑡)=𝑣_𝑚(𝑡), (24) or equivalently as presented in [24], 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵𝑣(𝑡), (25) where 𝑥=𝑥^𝑇𝑥_𝑛+1⋯𝑥_𝑛+𝑚^𝑇,𝐴(𝛼)=𝐴(𝛼)𝐵(𝛼)0_𝑚×𝑛0_𝑚×𝑚0,𝐵=^𝑛×𝑚𝐼_𝑚×𝑚. (26)

After the considerations above, note that the system (25) is similar to the system (17), and therefore the control problem falls into Case 1. Thus, one can adopt the procedure stated in Case 1 for designing a switched control law 𝑣(𝑡)=−𝐾_𝜎𝑥(𝑡),𝐾_𝜎∈ℝ^𝑛+𝑚.

5. Case 3: Linear System with Uncertainty in the Control Signal

In this case, it is assumed that the system (3) is the result of a linearization process of a plant 𝑥̇‌=𝑓(𝑥,𝑢), at an equilibrium point 𝑥=𝑥₀ and the respective control input 𝑢=𝑢₀. Suppose that 𝑥₀ is known, 𝑢₀ is uncertain because it depends on the plant uncertainties, but 0<𝑢₀∈[𝑢_0min,𝑢_0max] where 𝑢_0min and 𝑢_0max are known, and the linearized system is given by (2) and 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵(𝛼)𝑢(𝑡), (27) where 𝑥(𝑡)=𝑥(𝑡)−𝑥₀, 𝑥(𝑡) is the state vector of the plant;  𝑢(𝑡)=𝑢(𝑡)−𝑢₀, 𝑢(𝑡) is the control signal of the plant.

Now suppose that 𝐵(𝛼) can be written as follows: 𝐵(𝛼)=𝐵𝑔(𝛼), (28) where 𝐵 is a constant matrix and 𝑔(𝛼)>0, for all 𝛼 given in (2), is an bounded function that depends on uncertain parameters 𝛼. Thus, the system (27) can be written as follows: 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵(𝛼)𝑢(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵𝑔(𝛼)𝑢(𝑡). (29)

Assume that the gains 𝐾_𝑖=𝑀_𝑖𝑋⁻¹,𝑖∈𝕂_𝑟, and the matrix 𝑃=𝑋⁻¹, have been obtained using the vertices of the polytope of the system (27) in the LMIs (9) from Theorem 2. Now, given a constant 𝜉>0, define the control law 𝑢(𝑡)=𝑢_(𝜎,𝜉)(𝑡)=𝑢_(𝜎,𝜉)(𝑡)−𝑢₀,with𝑢_(𝜎,𝜉)(𝑡)=−𝐾_𝜎𝑥(𝑡)+𝛾_𝜉, (30) where 𝐾_𝜎∈𝐾₁,𝐾₂,…,𝐾_𝑟,𝜎=argmin_{𝑖∈𝕂𝑟}−𝑥^𝑇𝑃𝐵𝐾_𝑖𝑥,𝛾_𝜉=𝑢_0max,if𝑥^𝑇𝑢𝑃𝐵<−𝜉,_0min−𝑢_0max𝑥^𝑇𝑢𝑃𝐵+𝜉_0max+𝑢_0min(2𝜉)⁻¹,if𝑥^𝑇𝑢𝑃𝐵≤𝜉,_0min,if𝑥^𝑇𝑃𝐵>𝜉. (31)

Within this context, the following theorem is proposed.

Theorem 9. Suppose that the conditions from Theorem 2 hold, from the system (27) with the control law (6) and obtain 𝐾_𝑖=𝑀_𝑖𝑋⁻¹, 𝑖∈𝕂_𝑟, and 𝑃=𝑋⁻¹. Then, the switched control law (30) and (31) makes the system (27) uniform ultimate bounded.

Proof. Consider a quadratic Lyapunov candidate function 𝑉=𝑥^𝑇𝑃𝑥. Define 𝑉̇_𝑢𝛼 and 𝑉̇_{𝑢(𝜎,𝜉)}, the time derivatives of 𝑉 for the system (27), with the control laws (6) and (30), (31), respectively. Then, 𝑉̇_{𝑢(𝜎,𝜉)}=2𝑥^𝑇𝑃𝑥̇=2𝑥^𝑇𝑃𝐴(𝛼)𝑥+𝐵𝑔(𝛼)𝑢_(𝜎,𝜉)=2𝑥^𝑇𝑃𝐴(𝛼)(𝛼)𝐾(𝛼)(𝛼)×𝑥−𝐵𝑥+𝐵𝑔𝑢_(𝜎,𝜉)−𝑢₀(𝛼)𝑥=𝑉̇+𝐾_𝑢𝛼+2𝑥^𝑇𝑃𝐵𝑔(𝛼)−𝐾_𝜎𝑥+𝛾_𝜉−𝑢₀(𝛼)𝑥=𝑉̇+𝐾_𝑢𝛼+2𝑔(𝛼)min_{𝑖∈𝕂𝑟}−𝑥^𝑇𝑃𝐵𝐾_𝑖𝑥+2𝑔(𝛼)𝑥^𝑇𝑃𝐵𝛾_𝜉−𝑢₀+𝐾(𝛼)𝑥. (32) Remembering that 𝛼_𝑖≥0,𝑖∈𝕂_𝑟 and ∑^𝑟_𝑖=1𝛼_𝑖=1, 𝑔(𝛼)>0, 𝑔(𝛼)𝐵=𝐵(𝛼) and noting that min_{𝑖∈𝕂𝑟}{−𝑥^𝑇𝑃𝐵𝐾_𝑖𝑥}≤−𝑥^𝑇𝑃𝐵(∑^𝑟_𝑖=1𝛼_𝑖𝐾_𝑖)𝑥, from (32) 𝑉̇_{𝑢(𝜎,𝜉)}≤𝑉̇_𝑢𝛼−2𝑥^𝑇𝑃𝐵(𝛼)^𝑟_𝑖=1𝛼_𝑖𝐾_𝑖𝑥+2𝑔(𝛼)𝑥^𝑇𝑃𝐵𝛾_𝜉−𝑢₀(𝛼)𝑥=𝑉̇+𝐾_𝑢𝛼+2𝑔(𝛼)𝑥^𝑇𝑃𝐵𝛾_𝜉−𝑢₀. (33) Now, if |𝑥^𝑇𝑃𝐵|>𝜉, then from (31), 𝑔(𝛼)𝑥^𝑇𝑃𝐵(𝛾_𝜉−𝑢₀)≤0. Thus, from (33) 𝑉̇_{𝑢(𝜎,𝜉)}≤𝑉̇_𝑢𝛼<0 for 𝑥≠0, since the system (27) with the control law (6) is globally asymptotically stable. Otherwise, if |𝑥^𝑇𝑃𝐵|≤𝜉, one obtains from (31): 𝑉̇_{𝑢(𝜎,𝜉)}≤𝑉̇_𝑢𝛼+2𝑔_max𝑥^𝑇𝑃𝐵⋅𝛾_𝜉−𝑢₀≤−𝜖‖𝑥‖²+2𝑔_max𝛾_𝜉−𝑢₀𝜉≤−𝜖‖𝑥‖²+2𝑔_max𝛾_𝜉+𝑢₀𝜉≤−𝜖‖𝑥‖²+4𝑔_max⋅𝑢_0max‖𝑥‖⋅𝜉≤−𝜖²+𝜖₁, (34) where −𝜖 denotes the maximum eigenvalue of 𝑃(𝐴(𝛼)−𝐵(𝛼)𝐾(𝛼))+(𝐴(𝛼)−𝐵(𝛼)𝐾(𝛼))^𝑇𝑃, for all 𝛼 defined in (2), 𝑔_max=max{𝑔(𝛼)} and 𝜖₁=4𝑔_max⋅𝑢_0max⋅𝜉. Therefore, according to [25], the controlled system is uniformly ultimately bounded and the proof is concluded.

6. Examples

In this section, examples will be used to illustrate the three cases presented. The figures will show that the LMIs used to find the controller gains (Theorems 2 and 5 and Corollary 3) are more relaxed than the classical LMIs (Theorems 1 and 4). The solutions of the LMIs for the design of the gains of the controllers in the next examples were carried out using the modelling language YALMIP [18] with the solver SeDuMi [19].

Example  1

Case 1. Stability.

Consider the uncertain linear system given by (17), where 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)^𝑇,𝐴(𝛼)=𝑎11035𝑏101,010,𝐵= (35) with 𝑎≤𝑎≤13 and 𝑏≤𝑏≤−36, where 𝑎 and 𝑏 are constant uncertain parameters. Thus, the uncertain matrix 𝐴(𝛼) belongs to the polytope of vertices: 𝐴₁𝐴₂𝐴₃𝐴₄=𝑎11035𝑏010𝑎11035−360101311035𝑏010.1311035−36010 (36) For the feasibility study, it was considered that the parameters 𝑎 and 𝑏 belong to the intervals 0≤𝑎≤10 and −97≤𝑏≤−90, and 𝐵_𝑖=𝐵, 𝑖=1,2,3,4.

In this example, the idea is to show that, considering only the stability, the methodology presented in this paper (Corollary 3) can be more efficient than the methodology used in the literature (Theorem 1). For this, it is sufficient to note that the region of feasibility using Corollary 3 is greater than the region obtained by using the Theorem 1, as shown in Figure 1.

Figure 1 

Feasible regions using Theorem 1 (“+”) and the proposed method with Corollary 3 (“∘”).

Example  2

Case 2. Stability and constraint on the norm of the controller gains.

Consider the uncertain linear system given in (23), where 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)^𝑇,𝐴(𝛼)=,𝐵(𝛼)=𝑎1135−1𝑏, (37) with 𝑎≤𝑎≤0 and −1≤𝑏≤𝑏, where 𝑎 and 𝑏 are constant uncertain parameters. As the matrix 𝐵(𝛼) is uncertain, one defines a new state variable 𝑥₃(𝑡)=𝑢(𝑡)=∫𝑣(𝑡)𝑑𝑡. Thus, 𝑥̇₃(𝑡)=𝑢̇(𝑡)=𝑣(𝑡) and one obtains the following extended system 𝑥̇(𝑡)𝑥̇₃(𝑡)=𝐴(𝛼)𝐵(𝛼)0_1×20𝑥(𝑡)𝑥₃(𝑡)+0_2×11𝑣, (38) or equivalently, 𝑥̇(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵𝑣, (39) where 𝑥=𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)^𝑇,𝐴(𝛼)=𝑎1135−1𝑏001.000,𝐵= (40)

Therefore, the matrix 𝐴(𝛼) belongs to the polytope of vertices: 𝐴₁𝐴₂𝐴₃𝐴₄=𝑎1135−1𝑏000𝑎1135−1−100001135−1𝑏000.01135−1−1000 (41) For the feasibility study, it was considered that the parameters 𝑎 and 𝑏 belong to the intervals −2≤𝑎≤−0.5, −0.5≤𝑏≤1 and 𝐵_𝑖=𝐵, 𝑖=1,2,3,4.

In control problems, it is important to consider performance indices, for instance, restrictions on the norm of the controller gains. Thus, to find the regions of feasibility of the system, Theorem 1 and Corollary 3 were used together with Theorem 6 in order to ensure stability and constraint in the control input. For the constraint on the norm of the control input, 𝜂=1600 and 𝜂_𝑥=2 were fixed. From Figure 2, it can be observed that the proposed method is more flexible, since its feasible area is greater than that provided by Theorem 1.

Figure 2 

Feasible regions using Theorems 1 and 6 (“+”) and the proposed method with Corollary 3 and Theorem 6 (“∘”).

Example  3

Case 3. Stability, decay rate, and constraint on the norm of the controller gains.

Consider the uncertain linear system given in (3), where 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)^𝑇,𝐴(𝛼)=01(𝛼)=0𝑎−1,𝐵𝑏, (42) with −27≤𝑎≤𝑎 and −10≤𝑏≤𝑏, where 𝑎 and 𝑏 are constant uncertain parameters that belong to the intervals −20≤𝑎≤15 and −9≤𝑏≤−5,𝑢=𝑢−𝑢₀ and 𝑢₀>0 the uncertain reference signal. Note that 𝐵(𝛼) is not constant and can be written as follows: 𝐵(𝛼)=𝐵𝑔(𝛼),  with  𝑔(𝛼)>0, where 𝐵=[0−1]^𝑇 and 𝑔(𝛼)=−𝑏. Thus, the problem lies in the conditions of Case 3.

As in previous examples, the region of feasibility of the system will be found, and in this case, Theorems 4 and 5 will be used along with Theorem 6, thus ensuring decay rate and constraint on the norm of the controller gains. Therefore, the vertices of the polytope, 𝐴₁=𝐴₂,𝐴₃=𝐴₄, 𝐵₁=𝐵₃, and 𝐵₂=𝐵₄, are as follows: 𝐴_1,2=01𝑎−1,𝐴_3,4=,𝐵01−27−1_1,3=0𝑏,𝐵_2,4=0.−10 (43) For the simulation of the feasibility region was specified in Theorems 4 and 5 a decay rate greater than or equal to 3 and in Theorem 6, the constraint on the norm of the controller gains by adopting 𝜂=400 and 𝜂_𝑥=2. Thus, from Figure 3 one observes that the proposed method has a less conservative feasible region than that obtained with Theorem 4, which shows its flexibility.

Figure 3 

Feasible regions using Theorems 4 and 6 (“+”) and the proposed method with Theorems 5 and 6 (“∘”).

7. Switched Control Design of a Magnetic Levitator

To illustrate, the control of a magnetic levitator is designed, whose mathematical model [17, page 24] is given by: 𝑚𝑦̈=−𝑘𝑦̇+𝑚𝑔−𝜆𝜇𝑖²21+𝜇𝑦², (44) where 𝑚 is the mass of the ball;  𝑔=9.8m/s² is the gravity acceleration;  𝜇=2m⁻¹, 𝜆=0.460H, and 𝑘=0.001Ns/m are constant parameters of the levitator;  𝑖 is the electric current; and  𝑦 is the position of the ball.

Define the state variables 𝑥₁=𝑦 and 𝑥₂=𝑦̇. Then, (44) can be written as follows [26]: 𝑥̇‌₁=𝑥₂,𝑥̇₂=𝑔−𝑘𝑚𝑥₂−𝜆𝜇𝑖²2𝑚1+𝜇𝑥₁². (45)

Consider that during the required operation, [𝑥₁𝑥₂]^𝑇∈𝐷, where 𝐷=𝑥₁𝑥₂^𝑇∈ℝ²∶0≤𝑥₁≤0.15. (46)

The goal of the simulation is to design a controller that keeps the ball in a desired position 𝑦=𝑥₁=𝑦₀. Thus, the equilibrium point of the system (45) is 𝑥_𝑒=[𝑥_1𝑒𝑥_2𝑒]^𝑇=[𝑦₀0]^𝑇.

From the second equation 𝑥̇‌₂ in (45), observe that in the equilibrium point, 𝑥̇‌₂=0 and 𝑖=𝑖₀, where 𝑖₀=(2𝑚𝑔/𝜆𝜇)(1+𝜇𝑦₀)².

Linearizing the system (45) around the equilibrium point 𝑥=𝑥_𝑒, one has 𝑥̇₁𝑥̇₂=012𝑔𝜇1+𝜇𝑦₀−𝑘𝑚𝑥₁𝑥₂+0−2𝜆𝜇𝑔√𝑚1+𝜇𝑦₀𝑢, (47) where 𝑥₁=𝑥₁−𝑦₀,𝑥₂=𝑥₂,𝑢=𝑖−𝑖₀,⟹𝑥₁=𝑥₁+𝑦₀,𝑥₂=𝑥₂,𝑖=𝑢+𝑖₀,⟹𝑥̇₁=𝑥̇₁,𝑥̇₂=𝑥̇₂,𝑖=𝑢+2𝑚𝑔𝜆𝜇1+𝜇𝑦₀². (48)

Consider the constant position 𝑦₀ varying is in the range 0.04≤𝑦₀≤0.11 and that the mass 𝑚 is uncertain and belongs to the interval 0.02≤𝑚≤0.08, with 𝑥₁=𝑥₁−𝑦₀. Thus, from (46), the domain of the linear system (47) is 𝐷₂=𝑥₁𝑥₂^𝑇∈ℝ²∶−0.11≤𝑥₁≤0.11,0.04≤𝑦₀≤0.11,0.02≤𝑚≤0.08. (49)

Observe that the system (47) can be written as in (29); that is, 𝑥̇=𝐴(𝛼)𝑥+𝐵𝑔(𝛼)𝑢, (50) where 𝐵=[0−1]^𝑇 and 𝑔(𝛼)=𝑔(𝑚,𝑦₀)=2𝜆𝜇𝑔/√𝑚(1+𝜇𝑦₀). Note that 𝑔(𝛼)>0, for all 𝑚,𝑦₀∈𝐷₂.

Considering the domain 𝐷₂ and the presented parameters of the levitator, one has the following vertices of the polytope for the system (47): 𝐴₁=𝐴₂=,𝐴0136.2963−0.0125₃=𝐴₄=,𝐴0136.2963−0.05₅=𝐴₆=,𝐴0132.1311−0.0125₇=𝐴₈=,𝐵0132.1311−0.05₁=𝐵₃=𝐵₅=𝐵₇=0−27.8025^𝑇,𝐵₂=𝐵₄=𝐵₆=𝐵₈=0−12.3060^𝑇. (51) Initially, the feasibility region of the decay rate by the constraint on the controller gains is found, that is, the vertices of the polytope, given in (51), used in Theorems 4 and 5 with decay rate greater than or equal to 𝛽, varying in the range 0≤𝛽≤6.5, and the LMIs that guarantee the constraint on the controller gains, given in Theorem 6, with 0≤𝜂≤15 and 𝜂_𝑥=10. As seen in Figure 4, the proposed method (Theorems 5 and 6) is more flexible than the method presented in the literature (Theorems 4 and 6).

Figure 4 

Feasible regions using Theorems 4 and 6 (“+”) and the proposed method with Theorems 5 and 6 (“∘”).

Fixing the parameters related to decay rate in 𝛽=5 and constraint on the controller gains, 𝜂=7 and 𝜂_𝑥=10, the following gains and symmetric positive definite matrices were obtained: ,,𝐾𝐾=−8.9515−1.3708𝑃=7.42621.11921.11920.1852₁=,𝐾−12.9904−2.1058₂=,𝐾−14.0305−2.2765₃=,𝐾−13.0763−2.1208₄=,𝐾−13.9741−2.2677₅=,𝐾−13.9571−2.2647₆=,𝐾−15.6959−2.5472₇=,𝐾−14.2103−2.3061₈=,,−15.4792−2.5121𝑃=6.75711.09641.09640.1809 (52) with the classical method (Theorems 4 and 6) and the proposed method (Theorems 5 and 6), respectively.

As the controller gains have been found, to implement the control law (30) one must find the maximum and minimum values of 𝑢₀=𝑖₀ in the domain 𝐷₂ given in (49). As 𝑖₀=(2𝑚𝑔/𝜆𝜇)(1+𝜇𝑦₀)², it follows that max_{𝑚,𝑦0∈𝐷2}𝑖₀𝑚,𝑦₀=1.5927,min_{𝑚,𝑦0∈𝐷2}𝑖₀𝑚,𝑦₀=0.7050. (53) Thus, setting 𝜉=10⁻⁴ the control law (30) for the levitator is given by 𝑢_(𝜎,𝜉)(𝑡)=𝑖_(𝜎,𝜉)(𝑡)−𝑖₀,with𝑖_(𝜎,𝜉)(𝑡)=−𝐾_𝜎𝑥(𝑡)+𝛾_𝜉, (54) where 𝐾_𝑖,𝑖∈𝕂₈ are given in (52), 𝐾_𝜎∈𝐾₁,𝐾₂,𝐾₃,𝐾₄,𝐾₅,𝐾₆,𝐾₇,𝐾₈,𝜎=argmin_𝑖∈𝕂8−𝑥(𝑡)^𝑇𝑃𝐵𝐾_𝑖𝑥(𝑡,𝛾_𝜉)=1.5927,if𝑥^𝑇𝑃𝐵<−𝜉,−4438.723𝑥^𝑇𝑃𝐵+1.1488,if𝑥^𝑇𝑃𝐵≤𝜉,0.7050,if𝑥^𝑇𝑃𝐵>𝜉. (55)

For the numerical simulation, at 𝑡=0s it considered the initial condition 𝑥(0)=[0.050]^𝑇, 𝑚=0.08Kg, and 𝑦₀=0.09m. Thus, 𝑥(0)=𝑥(0)−[𝑦₀0]^𝑇=[−0.040]^𝑇. In 𝑡=1s, from Figure 5, the system is practically at the point 𝑥(1)=[𝑥₁(1)𝑥₂(1)]^𝑇=[0.090]^𝑇. After changing 𝑚 from 0.08Kg to 𝑚=0.04Kg and 𝑦₀ from 0.09m to 0.05m at 𝑡=2s, one can see that the system is practically at the point 𝑥(2)=[0.050]^𝑇, which will be the new initial condition. Finally, changing 𝑚 from 0.04Kg to 0.05Kg and 𝑦₀ from 0.05m to 0.08m for 𝑡≥2s. Thus, as seen in Figure 5, 𝑥(∞)=[0.080]. Figure 5 illustrates the system response.

(a)

(b)

(c)

Figure 5 

Position 𝑦(𝑡)=𝑥1(𝑡), velocity 𝑥2(𝑡), and electric current 𝑖(𝑡) of the controlled system, using the switched controller (30)-(31) (continuous line) and the classic controller (4) (dotted line), considering 𝑦0=0.09m and 𝑚=0.08Kg,   and  𝑦0=0.05m and 𝑚=0.04Kg, 𝑦0=0.08m and 𝑚=0.05Kg, for 𝑡∈[0,2), 𝑡∈[2,4), and 𝑡≥4s, respectively, with decay rate 𝛽=5, constraint on controller gains 𝜂=7 and 𝜂𝑥=10 and 𝜉=10−4.

Note that the control law given in (30) and (31) ensures that the system (47) is uniformly ultimate bounded, where 𝑦₀ and 𝑚 are uncertain parameters and belong to the intervals 0.04≤𝑦₀≤0.11 and 0.02≤𝑚≤0.08. Moreover, as can be seen in Figure 5, the new controller design presented in this paper is more efficient than the design commonly used in the literature. Note also that for certain values of the decay rate 𝛽 and the constraint on input 𝜂 such as 𝛽=6 and 𝜂=10, the controller (30) can be designed and the controller (4) cannot be designed, as shown in Figure 4.

Observe that the function 𝛾_𝜉, given in (31), is important to ensure the uniform ultimate boundedness of the system and smoothness of the control input. Note that when 𝜉 is equal to zero, the function 𝛾_𝜉 is a discontinuous function, and therefore the control input can also be discontinuous, as shown in Figure 6. Thus, the designer must choose 𝜉 according to the requirements.

(a)

(b)

(c)

Figure 6 

Position 𝑦(𝑡)=𝑥1(𝑡), velocity 𝑥2(𝑡), and electric current 𝑖(𝑡) of the controlled system, using the switched controller (30)-(31), considering 𝑦0=0.09m and 𝑚=0.08Kg,𝑦0=0.05m and 𝑚=0.04Kg,and𝑦0=0.08m and 𝑚=0.05Kg, for 𝑡∈[0,2), 𝑡∈[2,4) and 𝑡≥4s, respectively, with decay rate 𝛽=5, constraint on controller gains 𝜂=7 and 𝜂𝑥=10 and 𝜉=0.

8. Conclusions

This paper proposed a new switched control design method for some classes of linear systems with polytopic uncertainties. In the proposed controller, the gain is chosen by a switching law that returns the smallest time derivative value of the Lyapunov function.

The LMI used to find the gains are less conservative than that with only one state feedback gain [17], as seen in Figures 1 to 4. An application in the magnetic levitator control design illustrated the proposal. Thus, the authors believe that the proposed method can have useful practical applications on the control design of linear systems. Future studies on this subject include applications in power electronics [27].

Acknowledgments

The authors would like to thank the Brazilian agencies CAPES, CNPq, and FAPESP which have supported this research.