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

Authors: Wallysonn A. de Souza; Marcelo C. M. Teixeira; Máira P. A. Santim; Rodrigo Cardim; Edvaldo Assunção
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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:(1)𝕂r={1,2,…,r},x(t)=x,V(x(t))=V,∥x∥2=xTx,(2)(A,B,K)(α)=∑i=1r‍αi(Ai,Bi,Ki),withαi≥0,∑i=1r‍αi=1,α=[α1,α2,…,αr]T,
where r=2s and s is the number of uncertain parameters in the plant.


## 2. Linear Systems with Polytopic Uncertainties

Consider the linear system with polytopic uncertainties(3)x˙(t)=A(α)x(t)+B(α)u(t),
where x(t)∈ℝn is the state vector, u(t)∈ℝm is the control input, A(α) and B(α) as in (2), with Ai∈ℝn×n and Bi∈ℝn×m for i∈𝕂r.Assuming that all state variables are available for feedback, the control law largely used in the literature is given by [17]:
(4)u(t)=-Kx(t),
where K∈ℝm×n. Replacing (4) in (3), one obtains the feedback system
(5)x˙(t)=A(α)x(t)-B(α)Kx(t)=∑i=1r‍αi(Ai-BiK)x(t).Define a feedback control law with the state vector as(6)u(t)=uα(t)=-∑i=1r‍αiKix(t)=-K(α)x(t),
where Ki∈ℝm×n, i∈𝕂r. Considering (2) and from (6) and (3),
(7)x˙(t)=A(α)x(t)-B(α)K(α)x(t)=∑i=1r‍∑j=1r‍αiαj(Ai-BiKj)x(t).


### 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∈ℝm×n such that, for all i∈𝕂r,
(8)XAiT+AiX-BiM-MTBiT≺0.
If there exists such a solution, the controller gain is given by K=MX-1.Theorem 2.
The equilibrium pointx=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 Mi∈ℝm×n such that, for all i,j∈𝕂r, the following LMIs hold:
(9)XAiT+AiX-BiMi-MiTBiT≺0,(Ai+Aj)X+X(Ai+Aj)T-BiMj-BjMi-MiTBjT-MjTBiT⪯0,i<j.
If (9) are feasible, the controller gains are given by Ki=MiX-1, i∈𝕂r.Proof.
Consider a quadratic Lyapunov candidate functionV=xTPx. Thus, from (7) note that
(10)V˙=x˙TPx+xTPx˙=∑i=1r‍∑j=1r‍αiαjxT(Ai-BiKj)TPx+xTP∑i=1r‍∑j=1r‍αiαj(Ai-BiKj)x=xT[∑i=1r‍αi2(AiTP+PAi-KiTBiTP-PBiKi)]x+xT[∑i=1r-1‍∑j=1+ir‍αiαj(AiTP+PAi+AjTP+PAj-KjTBiTP-PBiKj-KiTBjTP-PBjKi)∑i=1r-1‍∑j=1+ir‍]x.
Now, αi≥0,i∈𝕂r and ∑i=1r‍αi=1. Then, from (10), V˙<0 (for x≠0) if for i,j∈𝕂r(11)AiTP+PAi-KiTBiTP-PBiKi≺0,AiTP+PAi+AjTP+PAj-KjTBiTP-PBiKj-KiTBjTP-PBjKi⪯0,i<j.
DefiningX=P-1, Mi=KiX and pre- and postmultiplying (11) by X, one obtains (9). The proof is concluded.Corollary 3.
IfB1=B2=⋯=Br=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 Mi∈ℝm×n, such that for all i∈𝕂r,
(12)XAiT+AiX-BMi-MiTBT≺0.
If (12) is feasible, the controller gains are given by Ki=MiX-1, i∈𝕂r.Proof.
It is similar to the proof of Theorem2, considering Bi=B, i∈𝕂r, and noting that now (7) can be rewritten as x˙(t)=∑i=1r‍αi(Ai-BKi)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
(13)limt→∞eβt∥x(t)∥2=0
holds for all trajectories x(t). As in [17, page 66], one can use a quadratic Lyapunov function V=xTPx to establish a lower bound β for the decay rate, considering the condition V˙≤-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∈ℝm×n such that, for all i∈𝕂r, the following LMIs are satisfied:
(14)XAiT+AiX-BiM-MTBiT+2βX≺0.
If there exists such a solution, the controller gain is given by K=MX-1.Proof.
It is similar to that of Theorem1, considering V˙≤-2βV [17].Theorem 5.
The equilibrium pointx=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 Mi∈ℝm×n such that, for all i,j∈𝕂r,
(15)XAiT+AiX-BiMi-MiTBiT+2βX≺0,(Ai+Aj)X+X(Ai+Aj)T-BiMj-BjMi-MiTBjT-MjTBiT+4βX⪯0,i<j.
If (15) are feasible, the gains are given by Ki=MiX-1,i∈𝕂r.Proof.
The proof is similar to that of Theorem2, considering V˙≤-2βV.One can constraint the norm of the controller gains by imposing restrictions onMi,i∈𝕂r, and X-1 as in [21]. Thus, given the constants η>0 and ηx>0, imposing that MiTMi≺ηIn,i∈𝕂r, and X-1≺ηxIn, 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 thatKiKiT≤ηηx2Im,i∈𝕂r is enforced if there exist constants η>0 and ηx>0, such that the LMIs from Theorems 2 or 5 (or 1 or 4 replacing Ki=K and also Mi=M), with the LMIs below hold:
(16)[ηxInInInX]≽0,[ηInMiTMiIm]≽0,i∈𝕂r.Proof.
The proof is similar to that presented in [21].


## 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∈ℝm×n such that, for all i∈𝕂r,
(8)XAiT+AiX-BiM-MTBiT≺0.
If there exists such a solution, the controller gain is given by K=MX-1.Theorem 2.
The equilibrium pointx=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 Mi∈ℝm×n such that, for all i,j∈𝕂r, the following LMIs hold:
(9)XAiT+AiX-BiMi-MiTBiT≺0,(Ai+Aj)X+X(Ai+Aj)T-BiMj-BjMi-MiTBjT-MjTBiT⪯0,i<j.
If (9) are feasible, the controller gains are given by Ki=MiX-1, i∈𝕂r.Proof.
Consider a quadratic Lyapunov candidate functionV=xTPx. Thus, from (7) note that
(10)V˙=x˙TPx+xTPx˙=∑i=1r‍∑j=1r‍αiαjxT(Ai-BiKj)TPx+xTP∑i=1r‍∑j=1r‍αiαj(Ai-BiKj)x=xT[∑i=1r‍αi2(AiTP+PAi-KiTBiTP-PBiKi)]x+xT[∑i=1r-1‍∑j=1+ir‍αiαj(AiTP+PAi+AjTP+PAj-KjTBiTP-PBiKj-KiTBjTP-PBjKi)∑i=1r-1‍∑j=1+ir‍]x.
Now, αi≥0,i∈𝕂r and ∑i=1r‍αi=1. Then, from (10), V˙<0 (for x≠0) if for i,j∈𝕂r(11)AiTP+PAi-KiTBiTP-PBiKi≺0,AiTP+PAi+AjTP+PAj-KjTBiTP-PBiKj-KiTBjTP-PBjKi⪯0,i<j.
DefiningX=P-1, Mi=KiX and pre- and postmultiplying (11) by X, one obtains (9). The proof is concluded.Corollary 3.
IfB1=B2=⋯=Br=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 Mi∈ℝm×n, such that for all i∈𝕂r,
(12)XAiT+AiX-BMi-MiTBT≺0.
If (12) is feasible, the controller gains are given by Ki=MiX-1, i∈𝕂r.Proof.
It is similar to the proof of Theorem2, considering Bi=B, i∈𝕂r, and noting that now (7) can be rewritten as x˙(t)=∑i=1r‍αi(Ai-BKi)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
(13)limt→∞eβt∥x(t)∥2=0
holds for all trajectories x(t). As in [17, page 66], one can use a quadratic Lyapunov function V=xTPx to establish a lower bound β for the decay rate, considering the condition V˙≤-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∈ℝm×n such that, for all i∈𝕂r, the following LMIs are satisfied:
(14)XAiT+AiX-BiM-MTBiT+2βX≺0.
If there exists such a solution, the controller gain is given by K=MX-1.Proof.
It is similar to that of Theorem1, considering V˙≤-2βV [17].Theorem 5.
The equilibrium pointx=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 Mi∈ℝm×n such that, for all i,j∈𝕂r,
(15)XAiT+AiX-BiMi-MiTBiT+2βX≺0,(Ai+Aj)X+X(Ai+Aj)T-BiMj-BjMi-MiTBjT-MjTBiT+4βX⪯0,i<j.
If (15) are feasible, the gains are given by Ki=MiX-1,i∈𝕂r.Proof.
The proof is similar to that of Theorem2, considering V˙≤-2βV.One can constraint the norm of the controller gains by imposing restrictions onMi,i∈𝕂r, and X-1 as in [21]. Thus, given the constants η>0 and ηx>0, imposing that MiTMi≺ηIn,i∈𝕂r, and X-1≺ηxIn, 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 thatKiKiT≤ηηx2Im,i∈𝕂r is enforced if there exist constants η>0 and ηx>0, such that the LMIs from Theorems 2 or 5 (or 1 or 4 replacing Ki=K and also Mi=M), with the LMIs below hold:
(16)[ηxInInInX]≽0,[ηInMiTMiIm]≽0,i∈𝕂r.Proof.
The proof is similar to that presented in [21].


## 3. Case 1: Linear Systems with a Constant MatrixB(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
(17)x˙(t)=A(α)x(t)+Bu(t).Suppose that (12) is feasible for all i∈𝕂r, and let Ki=MiX-1,i∈𝕂r, the gains of the controller given in (6), and P=X-1 obtained from the conditions of Corollary 3. Then, define the switched controller
(18)u(t)=uσ(t)=-Kσx(t),σ=argmini∈𝕂r{-x(t)TPBKix(t)},σ∈𝕂r.Note that, in (18) for a given t, the switching law σ can be obtained by calculating -x(t)TPBKix(t), for i∈𝕂r, and then considering the index set Ω(t)={j∈𝕂r:-x(t)TPBKjx(t)≤-x(t)TPBKix(t),∀i∈𝕂r}. 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
(19)x˙(t)=A(α)x(t)+Buσ(t)=∑i=1r‍αi(Ai-BKσ)x(t).Theorem 7.
Assume that the conditions of Corollary3, related to the system (17) with the control law (6), hold and obtain Ki=MiX-1, i∈𝕂r, 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 functionV=xTPx. Define V˙uα and V˙uσ, the derivatives of V for the system (17), with the control laws (6) and (18), respectively. Then, from (17) and (18),
(20)V˙uσ=2xTPx˙=2xTP(A(α)x+Buσ)=2xTPA(α)x+2xTPB(-Kσ)x.
From (2), ∑i=1r‍αi=1 and αi≥0,i∈𝕂r. Thus, note that
(21)mini∈𝕂r{xTPB(-Ki)x}≤xTPB(-∑i=1r‍αiKi)x,
and from (20), the switching law given in (18) and (6), observe that
(22)V˙uσ=2xTPA(α)x+2mini∈𝕂r{xTPB(-Ki)x}≤2xTPA(α)x+2xTPB(-∑i=1r‍αiKi)x=2xTP(A(α)-BK(α))x=2xTP(A(α)x+Buα)=V˙uα.
Therefore, V˙uσ(x(t))≤V˙uα(x(t)) and the proof is concluded.Remark 8.
Theorem7 shows that if the conditions of Corollary 3 are satisfied, then V˙uα(x(t))<0 for all x(t)≠0 and therefore V˙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,…,Kr 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 αi, i∈𝕂r, 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 MatrixB(α)

In this case, the linear system with polytopic uncertainties will be considered as given in (3); with αi,i∈𝕂r, defined in (2), namely,
(23)x^˙(t)=A^(α)x^(t)+B^(α)u(t),A^(α)=∑i=1r‍αiA^i,B^(α)=∑i=1r‍αiB^i.Letv(t)∈ℝm be the time derivative of the control input vector u(t)∈ℝm. Define xn+l(t) and vl(t), such that x˙n+l(t)=u˙l(t)=vl(t),l=1,2,…,m. Thus, one obtains the following system:
(24)x^˙(t)=A^(α)x^(t)+B^(α)u(t),x˙n+1(t)=v1(t),⋮x˙n+m(t)=vm(t),
or equivalently as presented in [24],
(25)x˙(t)=A(α)x(t)+Bv(t),
where
(26)x=[x^Txn+1⋯xn+m]T,A(α)=[A^(α)B^(α)0m×n0m×m],B=[0n×mIm×m].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)=-Kσx(t),Kσ∈ℝn+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 x-˙=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∈[u0min,u0max] where u0min and u0max are known, and the linearized system is given by (2) and
(27)x˙(t)=A(α)x(t)+B(α)u(t),
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 thatB(α) can be written as follows:
(28)B(α)=Bg(α),
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:
(29)x˙(t)=A(α)x(t)+B(α)u(t)=A(α)x(t)+Bg(α)u(t).Assume that the gainsKi=MiX-1,i∈𝕂r, 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
(30)u(t)=u(σ,ξ)(t)=u-(σ,ξ)(t)-u0,withu-(σ,ξ)(t)=-Kσx(t)+γξ,
where
(31)Kσ∈{K1,K2,…,Kr},σ=argmini∈𝕂r{-xTPBKix},γξ={u0max,ifxTPB<-ξ,((u0min-u0max)xTPB+ξ(u0max+u0min))(2ξ)-1,if|xTPB|≤ξ,u0min,ifxTPB>ξ.Within this context, the following theorem is proposed.Theorem 9.
Suppose that the conditions from Theorem2 hold, from the system (27) with the control law (6) and obtain Ki=MiX-1, i∈𝕂r, 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 functionV=xTPx. Define V˙uα and V˙u(σ,ξ), the time derivatives of V for the system (27), with the control laws (6) and (30), (31), respectively. Then,
(32)V˙u(σ,ξ)=2xTPx˙=2xTP(A(α)x+Bg(α)u(σ,ξ))=2xTP[(u-(σ,ξ)-u0+K(α)x)A(α)x-B(α)K(α)x+Bg(α)66666×(u-(σ,ξ)-u0+K(α)x)]=V˙uα+2xTPBg(α)(-Kσx+γξ-u0+K(α)x)=V˙uα+2g(α)mini∈𝕂r{-xTPBKix}+2g(α)xTPB[γξ-u0+K(α)x].
Remembering that αi≥0,i∈𝕂r and ∑i=1r‍αi=1, g(α)>0, g(α)B=B(α) and noting that mini∈𝕂r{-xTPBKix}≤-xTPB(∑i=1r‍αiKi)x, from (32)
(33)V˙u(σ,ξ)≤V˙uα-2xTPB(α)(∑i=1r‍αiKi)xV˙u(σ,ξ)≤+2g(α)xTPB[γξ-u0+K(α)x]V˙u(σ,ξ)=V˙uα+2g(α)xTPB(γξ-u0).
Now, if |xTPB|>ξ, then from (31), g(α)xTPB(γξ-u0)≤0. Thus, from (33) V˙u(σ,ξ)≤V˙uα<0 for x≠0, since the system (27) with the control law (6) is globally asymptotically stable. Otherwise, if |xTPB|≤ξ, one obtains from (31):
(34)V˙u(σ,ξ)≤V˙uα+2gmax|xTPB|·|γξ-u0|≤-ϵ∥x∥2+2gmax|γξ-u0|ξ≤-ϵ∥x∥2+2gmax(|γξ|+|u0|)ξ≤-ϵ∥x∥2+4gmax·u0max·ξ≤-ϵ∥x∥2+ϵ1,
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), gmax=max{g(α)} and ϵ1=4gmax·u0max·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 (Theorems2 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  1Case 1.
Stability.Consider the uncertain linear system given by (17), where
(35)x(t)=[x1(t)x2(t)x3(t)]T,A(α)=[a-11035b-010],B=[101],
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:
(36)[A1A2A3A4]=[a11035b010a11035-360101311035b0101311035-36010].
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 Bi=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 (Corollary3) 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 Theorem1 (“+”) and the proposed method with Corollary 3 (“∘”).Example  2Case 2.
Stability and constraint on the norm of the controller gains.Consider the uncertain linear system given in (23), where
(37)x^(t)=[x1(t)x2(t)]T,A^(α)=[a-135-1],B^(α)=[1b-],
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, x˙3(t)=u˙(t)=v(t) and one obtains the following extended system
(38)[x^˙(t)x˙3(t)]=[A^(α)B^(α)01×20][x^(t)x3(t)]+[02×11]v,
or equivalently,
(39)x˙(t)=A(α)x(t)+Bv,
where
(40)x=[x1(t)x2(t)x3(t)]T,A(α)=[a-1135-1b-000],B=[001].Therefore, the matrixA(α) belongs to the polytope of vertices:
(41)[A1A2A3A4]=[a1135-1b000a1135-1-100001135-1b00001135-1-1000].
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 Bi=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, Theorem1 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 ηx=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 Theorems1 and 6 (“+”) and the proposed method with Corollary 3 and Theorem 6 (“∘”).Example  3Case 3.
Stability, decay rate, and constraint on the norm of the controller gains.Consider the uncertain linear system given in (3), where
(42)x(t)=[x1(t)x2(t)]T,A(α)=[01a--1],B(α)=[0b-],
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, Theorems4 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:
(43)A1,2=[01a-1],A3,4=[01-27-1],B1,3=[0b],B2,4=[0-10].
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 ηx=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 Theorems4 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:
(44)my¨=-ky˙+mg-λμi22(1+μy)2,
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 variablesx-1=y and x-2=y˙. Then, (44) can be written as follows [26]:
(45)x-˙1=x-2,x-˙2=g-kmx-2-λμi22m(1+μx-1)2.Consider that during the required operation,[x-1x-2]T∈D, where
(46)D={[x-1x-2]T∈ℝ2:0≤x-1≤0.15}.The goal of the simulation is to design a controller that keeps the ball in a desired positiony=x-1=y0. Thus, the equilibrium point of the system (45) is x-e=[x-1ex-2e]T=[y00]T.From the second equationx-˙2 in (45), observe that in the equilibrium point, x-˙2=0 and i=i0, where i0=(2mg/λμ)(1+μy0)2.Linearizing the system (45) around the equilibrium point x-=x-e, one has
(47)[x˙1x˙2]=[012gμ1+μy0-km][x1x2]+[0-2λμgm(1+μy0)]u,
where
(48){x1=x-1-y0,x2=x-2,u=i-i0,⟹{x-1=x1+y0,x-2=x2,i=u+i0,⟹{x-˙1=x˙1,x-˙2=x˙2,i=u+2mgλμ(1+μy0)2.Consider the constant positiony0 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=x-1-y0. Thus, from (46), the domain of the linear system (47) is
(49)D2={[x1x2]T∈ℝ2:-0.11≤x1≤0.11,0.04≤y0≤0.11,0.02≤m≤0.08[x1x2]T}.Observe that the system (47) can be written as in (29); that is,
(50)x˙=A(α)x+Bg(α)u,
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 domainD2 and the presented parameters of the levitator, one has the following vertices of the polytope for the system (47):
(51)A1=A2=[0136.2963-0.0125],A3=A4=[0136.2963-0.05],A5=A6=[0132.1311-0.0125],A7=A8=[0132.1311-0.05],B1=B3=B5=B7=[0-27.8025]T,B2=B4=B6=B8=[0-12.3060]T.
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 ηx=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 Theorems4 and 6 (“+”) and the proposed method with Theorems 5 and 6 (“∘”).Fixing the parameters related to decay rate inβ=5 and constraint on the controller gains, η=7 and ηx=10, the following gains and symmetric positive definite matrices were obtained:
(52)K=[-8.9515-1.3708],P=[7.42621.11921.11920.1852],K1=[-12.9904-2.1058],K2=[-14.0305-2.2765],K3=[-13.0763-2.1208],K4=[-13.9741-2.2677],K5=[-13.9571-2.2647],K6=[-15.6959-2.5472],K7=[-14.2103-2.3061],K8=[-15.4792-2.5121],P=[6.75711.09641.09640.1809],
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
(53)maxm,y0∈D2{i0(m,y0)}=1.5927,minm,y0∈D2{i0(m,y0)}=0.7050.
Thus, setting ξ=10-4 the control law (30) for the levitator is given by
(54)u(σ,ξ)(t)=i(σ,ξ)(t)-i0,withi(σ,ξ)(t)=-Kσx(t)+γξ,
where Ki,i∈𝕂8 are given in (52),
(55)Kσ∈{K1,K2,K3,K4,K5,K6,K7,K8},σ=argmini∈𝕂8{-x(t)TPBKix(t)},γξ={1.5927,ifxTPB<-ξ,-4438.723xTPB+1.1488,if|xTPB|≤ξ,0.7050,ifxTPB>ξ.For the numerical simulation, att=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)=[x-1(1)x-2(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.Positiony(t)=x-1(t), velocity x-2(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.
(a)
(b)
(c)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 functionγξ, 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 γξ 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.Positiony(t)=x-1(t), velocity x-2(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.
(a)
(b)
(c)


## 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].



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