# Synthesis of Adaptive Gain Robust Controllers for Polytopic Uncertain Systems

Authors: Hidetoshi Oya; Daisuke Yamasaki; Shunya Nagai; Kojiro Hagino
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We present a new adaptive gain robust controller for
polytopic uncertain systems.
The proposed adaptive gain robust controller consists of a state
feedback law with a fixed gain and a compensation input with adaptive
gains which are tuned by updating laws.
In this paper, we show that sufficient conditions for the existence of
the proposed adaptive gain robust controller are given in terms of LMIs.
Finally, illustrative examples are presented to show the effectiveness
of the proposed adaptive gain robust controller.


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## Body

## 1. Introduction

In order to design control systems, it is necessary to derive a mathematical model of the controlled system. It is well known that if the mathematical model describes the controlled system completely, one can design a satisfactory control system by using some controller design methods. However, there are some gaps between the controlled system and its mathematical model and the gaps are referred to as uncertainties. The uncertainties included in the mathematical model may occur deterioration of control performance and, at the worst, control systems become unstable. Therefore, robust stability analysis and robust stabilization for uncertain dynamical systems have received much attention for a long time (e.g., see [1] and references therein). In particular, there are lots of existing results for state feedback robust control such as quadratic stabilizing controllers, H∞ control systems (e.g., see [2–4]), and so on. In addition for a class of uncertain linear systems a connection between H∞ control and quadratic stabilization has been established [5].By the way in most practical situations, it is desirable to design robust control systems which achieve not only robust stability but also an adequate level of performance. Thus, robust controllers with achievable performance level have also been well studied. For instance, guaranteed cost control is introduced by Chang and Peng [6], and Riccati equation approach [7] and LMI one [8] have been suggested. Additionally, several researches have dealt with the problem of designing a robust controller that satisfies additional constraints on the closed-loop poles’ location (e.g., see [9, 10]). Besides, some design methods of variable gain controllers for uncertain linear systems have also been shown (e.g., see [11–13]). Maki and Hagino [11] have presented a robust controller with adaptation mechanism for linear systems with time-varying parameter uncertainties. In their work, the control gain is tuned online based on the information about parameter uncertainties and a target model with adjustable parameters is also introduced. In the work of Oya and Hagino [12] an error signal between the desired trajectory and the actual response is defined and an adaptive compensation input is determined so as to reduce the effect of uncertainties. These robust controllers consist of a fixed gain controller and a variable gain one, and the variable gain controller is adjusted by updating rules.In this paper, we consider a design problem of an adaptive gain robust controller for polytopic uncertain systems. The proposed adaptive gain robust controller is composed of a state feedback law with a fixed gain and a compensation input. The compensation input is defined as a state feedback with a fixed gain and one with adaptive gains tuned by updating rules. The advantage of our new adaptive gain robust control is that, for the case that conventional quadratic stabilizing controller based on Lyapunov criterion cannot be obtained, the proposed design method may be able to design stabilizing controller. This paper is organized as follows. In Section2, we show notations which are used in this paper, and Section 3 contains our main result. We show that sufficient conditions for the existence of the proposed robust controller are given in terms of LMIs. Finally, simple numerical examples are included to show the effectiveness of the proposed robust controller design approach.


## 2. Notations

In this section, we introduce notations which are used in this paper as well as the existing works (e.g., see [13]).In the paper, the following notations are used. For a matrixA, the transpose of the matrix A and its inverse are denoted by AT and A-1, respectively. Additionally He{A} and In mean A+AT and n-dimensional identity matrix, respectively, and the notation diag⁡(A1,…,AM) represents a block diagonal matrix composed of matrices Ai for i=1,…,M. For real symmetric matrices A and B, A>B(resp., A≥B) means that A-B is positive (resp., nonnegative) definite matrix. For a vector α∈Rn, α denotes the standard Euclidian norm and, for a matrix A, A represents its induced norm. The symbols “≜” and “⋆” mean equality by definition and symmetric blocks or symmetric elements in matrices, respectively.


## 3. Problem Formulation

Consider the following uncertain linear system:(1)ddtxt=A+∑k=1N‍θkDkxt+But,where x(t)∈Rn and u(t)∈Rm are the vectors of state (assumed to be available for feedback) and the control input, respectively. In (1), the matrices A∈Rn×n and B∈Rn×m are the nominal values of system parameters, and the matrix Dk∈Rn×n(k=1,…,N) denotes the structure of uncertainties. Besides, θk∈R1 represents unknown parameters which satisfy the relation (2)∑k=1N‍θk=1,θk≥0.Namely, the uncertain system of (1) is a class of polytopic uncertain systems. Additionally, we assume that the pairs (A,B) and (A+Dk,B) are stabilizable.The nominal system, ignoring the unknown parameters in (1), is given by (3)ddtx¯t=Ax¯t+Bu¯t.In this paper, first of all, we consider the standard linear quadratic control problem for the nominal system of (3). Namely, we define the following quadratic cost function for the nominal system of (3):(4)I=∫0∞‍x¯TtQx¯t+u¯TtRu¯tdt,where the matrices Q∈Rn×n and R∈Rm×m which are selected by designers are semipositive and positive definite. It is well-known that the optimal control input minimizing the quadratic cost function of (4) is given by u¯(t)=Kx¯(t), where K∈Rm×n represents the optimal control gain matrix for the nominal system of (3). Note that the closed-loop system matrix AK≜A+BK is stable (the design parameter Q∈Rn×n is selected such that the pair (A,H) is detectable, where H is any matrix satisfying Q=HHT, and then the closed-loop system matrix AK is stable) and the optimal feedback gain matrix K∈Rm×n can be computed as K=-R-1BTP, where P∈Rn×n is the unique solution of the algebraic Riccati equation: (5)HeATP-PBR-1BTP+Q=0.Now, by using the optimal feedback gain matrixK∈Rm×n for the nominal system of (3), we consider the following control input: (6)ut≜Kxt+ξx,θ^k,t,where ξ(x,θ^k,t)∈Rm is a compensation input so as to compensate the effect of unknown parameters [12]. The compensation input ξ(x,θ^k,t) is defined as (7)ξx,θ^k,t≜Kxt+∑k=1N‍θ^ktKkxt,where K∈Rm×n and Kk∈Rm×n(k=1,…,N) are fixed gain matrices and θ^k(t)(k=1,…,N) are time-varying adjustable parameters. The decision method of fixed gain matrices K and Kk and adjustable parameters θ^k(t) are shown in Section 4. From (1), (6), and (7), we have the uncertain closed-loop system described as(8)ddtxt=A+∑k=1N‍θkDkxt+BKxt+Kxt+∑k=1N‍θ^kKktxt=AK+BK+∑k=1N‍θkDkxt+∑k=1N‍θ^ktBKkxt.By introducing the complementary variables θ~k(t)≜θ^k(t)-θk(k=1,⋯,N), the uncertain closed-loop system of (8) can be rewritten as(9)ddtxt=∑k=1N‍θkAK+Dk+BKxt+∑k=1N‍θ~kt+θkBKkxt=∑k=1N‍θkAK+Dk+BKk+BKxt+∑k=1N‍θ~ktBKkxt.Here we have used the relation of (2). In addition, defining the time-varying matrix (10)ΓKx,t≜BK1xt,BK2xt,…,BKNxt,we obtain the following uncertain closed-loop system: (11)ddtxt=∑k=1N‍θkADk+BKxt+ΓKx,tθ~t,where ADk and θ~(t) are given by (12)ADk=AK+Dk+BKk,θ~t=θ~1t,…,θ~NtT.From the above, our control objective in this paper is to design the compensation input of (7) such that the uncertain closed-loop system of (11) is asymptotically stable, that is, deriving the fixed gain matrices K∈Rm×n and Kk∈Rm×n and updating rules of adjustable parameters θ^k(t) which stabilize the uncertain closed-loop system of (11).


## 4. Main Results

In this section, we show a design method of the proposed adaptive gain robust controller such that the uncertain closed-loop system of (11) is asymptotically stable. The following theorem shows a sufficient condition for the existence of the proposed robust control system.Theorem 1.
Consider the uncertain linear system of (1) and the control input of (6).
Firstly, in order to design the fixed gain matricesKk∈Rm×n we solve the following LMIs for the symmetric positive definite matrix Xk∈Rn×n and the matrix Yk∈Rm×n:(13)HeAK+DkXk+BYk+Qk<0,where Qk∈Rn×n(k=1,…,N) are symmetric positive definite matrices selected by designers. If there exist the symmetric positive definite matrix Xk∈Rn×n and the matrix Yk∈Rm×n satisfying the LMIs of (13), the fixed gain matrices Kk∈Rm×n are determined as Kk=YkXk-1.
Next, the fixed gain matrixK∈Rm×n and the updating rule of adjustable parameters θ^k(t)∈RN are designed as (14)K=WS-1,ddtθ^t=-Σθ-1ΓKTx,tS-1xtprovided that the following LMIs for the symmetric positive definite matrix S∈Rn×n and the matrix W∈Rm×n are feasible:(15)HeADkS+BW<0k=1,…,N,where Σθ∈RN×N in (14) is the symmetric positive definite matrix selected by designers. Then asymptotical stability of the uncertain closed-loop system of (11) is guaranteed.Proof.
First of all, for the design parametersQk∈Rn×n if the LMIs of (13) are feasible, then, by solving the LMIs of (13), the fixed gain matrices Kk∈Rm×n are derived as Kk=YkXk-1. Note that then the matrices ADk+BK(k=1,…,N) are stable.
Next, we introduce the following quadratic function as a Lyapunov function candidate:(16)Vx,t≜xTtXxt+θ~TtΣθθ~t,where X∈Rn×n and Σθ∈RN×N are symmetric positive definite matrices. Note that the matrix X∈Rn×n satisfies X=S-1. Namely, the quadratic function V(x,t) is positive definite and radially unbounded. The time derivative of the quadratic function V(x,t) along the trajectory of the uncertain closed-loop system of (11) can be expressed as:(17)ddtVx,t=xTtHe∑k=1N‍θkXADk+BKxt+HexTtXΓKx,tθ~t+ddtθ~tTΣθθ~t+θ~TtΣθddtθ~t.
Thus from the updating rule of (14) and the relation of (17) we have (18)ddtVx,t=xTtHe∑k=1N‍θkXADk+BKxt.Namely, if the matrix inequality (19)He∑k=1N‍θkXADk+BK<0is satisfied, then the following relation for ∀x(t)≠0 holds:(20)ddtVx,t<0;that is, the uncertain closed-loop system of (11) is asymptotically stable.
By introducing the matrixW=KS one can easily see that the condition of (19) is equivalent to (21)He∑k=1N‍θkADkS+BW<0.Since the unknown parameters θk satisfy the relation of (2), we find that the inequality condition of (21) is equivalent to the LMIs of (15).
Thus the proof of Theorem1 is accomplished.Remark 2.
It is well known that if the LMIs of (A.1) are not feasible then the conventional quadratic stabilizing controller cannot be designed. Furthermore, the conventional design method does not have design parameters. On the other hand, in this paper, by introducing the adjustable parameter, the control input is constructed as (6) and (7). One can see from stabilizability of the pair (A+Dk,B) that the LMIs of (13) are feasible. Namely, the fixed gain matrices Kk∈Rm×n can always be derived and the matrix ADk=AK+Dk+BKk in (12) is stable, because the fixed gain matrices Kk are derived by using the solution of LMIs of (13). Besides, LMIs of (15) may be feasible provided that the LMIs of (A.1) are not feasible (see Section 5). Therefore, for the case that the conventional quadratic stabilizing controller cannot be derived, the proposed design approach has possibility of robust stabilizing controller design. Note that the analysis of conservativeness for the proposed robust controller is one of our future research subjects.Remark 3.
In this paper, by introducing adjustable parametersθ^k(t)(k=1,…,N), we have proposed an adaptive gain robust controller. The adaptive action of the time-varying parameters θ^k(t) can be adjusted by selecting the design parameter Σθ∈RN×N in (14). In order to construct an adaptive robust control system, the adjustable parameters θ^k(t)(k=1,…,N) are introduced and the time-varying adjustable parameter θ^k(t) differs from an estimate for the unknown parameter. Namely, we introduce the adjustable parameter θ^k(t) so as to reduce the effect for the uncertainties adaptively and determining θ^(t) in (14) is not to estimate the unknown parameter θk [14]. Besides, the value of the adjustable parameter θ^k(t) depends on its initial value and the design parameter Σθ∈RN×N in (14) and the adjustable parameters θ^k(t) do not satisfy the relation of (2) usually (see Example 1 in Section 5).Remark 4.
In this paper, we consider the polytopic uncertain system of (1) which has uncertainties in the state matrix only. The proposed design method of the adaptive gain robust controller can also be applied to the case that the uncertainties are included in both the system matrix and the input one. By introducing additional actuator dynamics and constituting an augmented system, the uncertainties in the input matrix are embedded in the system matrix of the augmented system [15]. Therefore the same design procedure can be applied.


## 5. Numerical Examples

In order to demonstrate the efficiency of the proposed control scheme, we have run two simple numerical examples. The control problems considered here are not necessarily practical. However, the simulation results stated below illustrate the distinct feature of the proposed controller design.Example 1.
Consider the uncertain linear system(22)ddtxt=-2.0+δ11.00.01.0+δ2xt+0.01.0ut,where δ1 and δ2 are unknown parameters and these parameters are supposed to vary within the intervals -1.5,1.5 and -1.0,1.0, respectively. Namely, one can see that the matrices Dk(k=1,…,4) in (1) can be described as (23)D1=-1.50.0⋆-1.0,D2=1.50.0⋆-1.0,D3=-1.50.0⋆1.0,D4=1.50.0⋆1.0.Now we select the weighting matrices Q∈R2×2 and R∈R1×1 such as Q=1.0×I2 and R=3.0 in (5). Then, solving the algebraic Riccati equation of (5), we obtain (24)K=-1.8142×10-1-3.0887×10-1,P=7.4177×10-11.8142×10-1⋆3.0887.Next, we solve LMIs of (13). The design parameters Qk∈R2×2 in (13) are set at Q1=1.0×I2, Q2=3.0×I2, Q3=1.0×101×I2, and Q4=7.0×I2 and then we can obtain symmetric positive definite matrices Xk∈R2×2 such as (25)X1=1.8171×1031.3633×102⋆2.7714×103,X2=2.6483×1038.2403×10-1⋆2.6475×103,X3=1.8185×1031.3610×102⋆2.7712×103,X4=2.6497×103-2.1580⋆2.6475×103and matrices Yk∈R1×2 given by (26)Y1=-1.54407.2608×103,Y2=-2.17016.8521×103,Y3=-1.81871.7131×103,Y4=-2.17041.5548×103.Thus we have the following gain matrices(27)K1=-1.05012.6715,K2=-8.1862×10-12.5879,K3=-1.05026.6976×10-1,K4=-8.18665.8659×10-1.Besides, solving the LMIs of (15), we obtain (28)S=1.3942-1.0158⋆4.3169,W=-2.9440-1.3793,K=-2.8294-9.8528×10-1.From the above, the proposed adaptive gain robust controller can be obtained.
Now, in this example, the initial value for the uncertain system of (22) and a design parameter for the updating law of (14) are selected as x(0)=(1.0-2.0)T and Σθ=1.0×101×I4, respectively, and the initial value of the adjustable parameter is set at θ^(0)=(0.00.00.00.0)T. Besides, we consider the following two cases for the unknown parameters δk(k=1,2): (i)
Case  1:δ1=-1.5 and δ2=-1.0;(ii)
Case  2:δ1=-1.5 and δ2=1.0;that is, the unknown parameters θk are given as follows. (i)
Case  1:θ1=1.0,θ2=θ3=θ4=0.0.(ii)
Case  2:θ1=θ2=θ4=0.0,θ3=1.0.
The results of the simulation of this example are depicted in Figures1–5. In Figures 1–3, “Case  1” and “Case  2” represent the time histories of the state variables x1(t) and x2(t) and the control input u(t). Figures 4 and 5 show the time histories of the adjustable parameters θ^k(t)(k=1,…,4).
From Figures1–5, we find that the proposed adaptive gain robust controller stabilizes the linear system of (22) in spite of plant uncertainties. In addition, one can see from Figures 4 and 5 that in this example the parameters θ^1(t) and θ^2(t) take negative value; that is, θ^k(t) does not satisfy the relation of (2) and the time-varying adjustable parameter θ^k(t) does not mean an estimate of unknown parameter θk∈R1.Figure 1
Time histories of the statex1(t).Figure 2
Time histories of the statex2(t).Figure 3
Time histories of the control inputu(t).Figure 4
Time histories of adjustable parametersθ^k(t): Case 1.Figure 5
Time histories of adjustable parametersθ^k(t): Case  2.Example 2.
Consider the linear system with coefficient matrices of(29)A=1.51.00.00.01.02.0-2.0-1.0-1.0,B=0.00.01.0,D1=0.0-9.5×10-10.00.00.00.00.00.00.0,D2=0.09.5×10-10.00.00.00.00.00.00.0,D3=0.00.00.00.00.0-1.5×10-10.00.00.0,D4=0.00.00.00.00.01.5×10-10.00.00.0.By selecting the design parameters in (13) such as Q=3.0×I3, R=4.0, Q1=3.0×I3, Q2=9.0×I3, Q3=1.0×101×I3, and Q4=8.0×I3, respectively, and solving the algebraic Riccati equation of (5), we have(30)K=-1.1174×101-9.5394-5.3173,P=2.2507×1021.2893×1024.4696×101⋆8.9733×1013.8158×101⋆⋆2.1269×101.Additionally, the solution of LMIs of (13) and fixed gain matrices Kk∈R1×3(K=1,…,4) can be computed as(31)X1=1.1711×101-4.0285×1029.3258×101⋆1.8472×104-1.1168×104⋆⋆2.5536×104,Y1=-3.0832×103-7.8885×1043.8937×104,X2=9.1439×103-8.3256×1031.7793×102⋆1.1592×104-7.5956×103⋆⋆2.5293×104,Y2=4.8391-7.84887.6165×104,X3=1.0718×103-1.8219×1032.6448×103⋆3.7569×103-8.8775×103⋆⋆4.7077×104,Y3=1.6538×104-5.5147×1042.3278×105,X4=4.2455×103-7.737×103-1.8843×102⋆2.1156×104-7.4184×103⋆⋆1.9875×104,Y4=1.90741.20193.9039×104,(32)K1=-3.8863×102-3.29651.5024,K2=-8.6679×10-1-6.74309.9241×10-1,K3=-4.4573×101-3.3716×1011.0909,K4=-7.9847-1.94381.1646.Finally, by using these gain parameters of (32), we can obtain the following solution of the LMIs of (15) and a fixed gain matrix K∈R1×3 of (34):(33)S=5.4616×10-4-1.6403×10-24.7503×10-3⋆8.9260×10-1-2.5601×101⋆⋆2.3132×103,W=9.2471×10-19.1270×10-21.400110-5.Thus the following gain matrix can be computed:(34)K=-6.0417×105-2.0255×104-2.8346×102.
On the other hand, the uncertain system with the coefficient matrices of (29), the conventional quadratic stabilizing controller based on Lyapunov criterion, cannot be designed; that is, the LMIs of (A.1) are not feasible. This result shows that, for the case such that the robust controller based on Lyapunov criterion cannot be obtained, the proposed controller design method has possibility of robust control system design. Therefore, the proposed design approach is very useful.


## 6. Conclusions

This paper has dealt with a design problem of adaptive gain robust controllers for polytopic uncertain systems. The proposed adaptive gain robust controller consists of fixed gain parameters and time-varying adjustable ones. In this paper, we show an LMI-based design algorithm of the proposed robust controller and simple numerical examples are given for illustration of the proposed controller design method. The simulation result has shown that the closed-loop system is well stabilized in spite of plant uncertainties for the case such that the robust stabilizing controller based on quadratic stabilization cannot be derived.The proposed adaptive gain robust controller can easily be obtained by solving LMIs; that is, the proposed design method is very simple. Besides, one can see that, for the case that conventional quadratic stabilizing controller based on Lyapunov criterion cannot be obtained, the proposed controller design method has possibility of robust controller design. Therefore, the proposed design approach is very useful.The future research subjects are extensions of the proposed adaptive gain robust controller to such a broad class of systems as uncertain time-delay systems, uncertain discrete-time systems, and so on.



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