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*versión impresa* ISSN 0327-0793*versión On-line* ISSN 1851-8796

### Lat. Am. appl. res. vol.44 no.4 Bahía Blanca oct. 2014

**Active disturbance rejection control tuning employing the lqr approach for decoupling uncertain multivariable systems**

**P. Teppa-garran ^{†} and G.Garcia^{‡}**

^{†}* Departamento de Procesos y Sistemas, Universidad Simón Bolívar, Valle de Sartenejas, Municipio Baruta, Estado Miranda, Caracas, Venezuela. pteppa@usb.ve*

^{‡}* CNRS, LAAS, 7 Avenue du Colonel Roche, F-31400 Toulouse, France and Université de Toulouse, LAAS, INSA, Toulouse, France. garcia@laas.fr*

*Abstract*— The ADRC tuning is essentially a pole-placement technique and the desired performance is indirectly achieved through the location of the closed-loop poles. However, the final choice of these poles becomes a trial-and-error strategy. In contrast with pole-placement, in the LQR method, the desired performance objectives are directly and globally addressed by minimizing a quadratic function of the state and control input. ADRC tuning employing the LQR approach is then applied for decoupling uncertain MIMO systems. This is done by considering all the coupling and interference interactions between the channels of the system as disturbances, using an ESO to estimate them in real time and then canceling its effect employing the estimate as part of the control signal.

*Keywords*— Active Disturbance Rejection Control (ADRC); Extended State Observer (ESO); Linear Quadratic Regulator (LQR); Multiple-Input-Multiple-Output (MIMO) Systems; Decoupling.

**I. INTRODUCTION**

In many industrial plants, the basic extension of classical PID controller design, implementation and tuning is the decentralized approach, where structural concepts are used to decouple the interaction between variables. The control effort is decomposed into two stages: first to decouple the different subsystems and then to control them. Decoupling or non-interacting control is a popular approach to dealing with control loop interactions. Here, the objective is to eliminate completely the effects of loop interactions. Decoupling control was initially developed for deterministic linear systems. Typical approaches include design of state feedback to reach de-coupling of state equation (Falb and Wolovich, 1967), decoupling in frequency domain through inverse Nyquist array (Rosenbrock, 1969), decoupling via relative gain array (Bristol, 1966, Shinskey, 1979, Friedly, 1984), decoupling using Singular Value Decomposition of the transfer function matrix (Lau *et al*., 1989), and designing precompensators that transforms the controlled transfer function matrix into a diagonal matrix or diagonal dominance (Niederliski, 1981, Albertos and Sala*, *2004, Skogestad and Postlethwaite, 2005), where the precompensator can take the form of: dynamic de-coupling, steady state decoupling or decoupling at one particular frequency. All these different approaches separate the controlled multivariable system into several Single-Input-Single-Output (SISO) subsystems through a suitable decoupler that depends on accurate process model before controller design. So they are difficult to reach decoupling control of complex industrial multivariable processes characterized with strongly interactions and uncertainties. One of the main issues in control is to deal with uncertainties including internal (parameter and unmodeled dynamics) and external (disturbances). However, most uncertainties are not measurable. The extended state observer (ESO) (Han, 1998, 1999, Gao *et al*., 2001) has been proposed to estimate mixed uncertainties for nonlinear systems.

Active Disturbance Rejection Control (ADRC) (Han, 1998, 1999, 2009; Gao *et al., *2001) is a robust control method that does not require a detailed mathematical description of the system. It is based on the extension of the system model trough a virtual state variable, representing everything that it is not included in the mathematical model of the plant. An estimate of this state provided by an ESO can be further used in the control signal to decouple the real perturbation in the plant. It is this inherent capacity of decoupling of the ADRC method that has been employed in the control of MIMO systems. This is done by considering all the coupling and interference interactions between the channels of the system as disturbances, use an ESO to estimate them in real time and then canceling its effect employing the estimate as part of the control signal. This strategy has been used in the control of particular MIMO problems by decomposing the global system into several SISO subsystems and then designing ADRC for each loop, for example: (Huang *et al*., 2004, Liu *et al*., 2008, Khani and Yazdizadeh, 2009, Shi *et al*., 2012, Tian *et al*., 2012), to cite few of them. There are also some contributions that propose a general ADRC framework to treat the MIMO systems, we can mention in this case: (Xia *et al*., 2007) where MIMO systems with time delay are considered by viewing the system with time delay in the input as a high-order system without time delay in the input, (Miklosevic and Gao, 2005) where it is employed a dynamic decoupling method in the control of a performance turbofan engine and (Zheng *et al*., 2009) where a dynamic decoupling control based on SISOADRC is used for uncertain square MIMO systems with predetermined input-output pairs.

The tuning procedure in ADRC was originally proposed in a nonlinear form (Han, 1998, 1999, 2009; Gao *et al*., 2001), but the large number of gains made tuning an art. The structure was simplified to its linear form (Gao, 2003) and parameterized into a few gains. In its linear form, the tuning is essentially a pole-placement technique and the desired performance is indirectly achieved through the location of the closed-loop poles (controller and ESO). However, the final choice of these poles becomes a trial-and-error strategy. Linear Quadratic Regulator (LQR) (Kwakernaak and Sivan, 1972; Anderson and Moore, 1989) is a well-known design technique in modern optimal control theory and has been widely used in many applications. In contrast with pole-placement, the desired performance objectives are directly addressed by minimizing a quadratic function of the state and control input. The resulting optimal control law has many nice properties, including that of closed-loop stability. Furthermore, by the choice of the weighting matrices ** Q **and

**it is possible to control the tradeoff between the requirements of regulating the state and the expenditure of control energy.**

*R*In this paper, we propose a LQR solution to develop optimal tuning algorithms for decoupling uncertain MIMO systems that have been formulated into the ADRC framework. The method allows computing the gain matrices of the controller and the ESO directly and by considering the system in a global way avoiding the general standard approach of SISO-ADRC design into the MIMO case.

**II. MIMO-ADRC TUNING VIA THE LQR METHOD**

Let us consider a MIMO nonlinear time varying plant which can be described exactly in its operating range by the followings implicit coupled input-output equations

(1) |

where *S _{i}*(.) for

*i*=1,...,

*p*is a sufficiently smooth function of the external vector disturbance , and the vectors

(2) | |

(3) |

where is the control vector and is the controlled output. Assume that for some integer *n _{i},* such that 0<

*n*≤

_{i}*n*we have . The implicit function theorem yields then locally

_{yi},(4) |

with

Doing in (2), being a real unknown scaling vector of the system that can be approximated by the vector , we have

(5) |

where is the input disturbance that represents any difference between the model and the real system. That is, *f _{i} *includes the combined effects of unmodeled dynamics, external disturbances and loop interactions. We set

*n=n*

_{1}

*=n*

_{2}

*= ...= n*in (5). That is

_{p}(6) |

Defining the input matrix of the system , the generalized perturbation vector and the *n*-th derivative order output vector , we express (6) in compact form as

(7) |

By setting , we can remove the scaling matrix from (7), the plant equation changes now to

(8) |

The real control input is

(9) |

Here it is supposed that *p*≤*m *and so that the right inverse of matrix in (9) exists.

**A. MIMO-ADRC formulation **

Let the state vector be

The state space model of (8) can be written as

(10) |

where

Here * O_{p}* and

*are zero and identity matrices. Let the vector*

**I**_{p}

*x*

_{n}_{+1}=

**, the state vector is now and the state space model of the plant becomes**

*f*(11) | |

For system (11) the ESO is designed as follows

(12) |

where is the estimated of the state vector ** x**(

*t*)

*and the generalized perturbation*

*, and is the observer gain matrix with (computation of matrix*

**f****will be considered in section II.D). As the vector , it is used to actively cancel**

*L***(.) in (8) by applying**

*f*(13) |

This control law decouples the system into a set of parallel *n *integrator systems

(14) |

The same can be observed in the state space representation if we replace (13) in (10) resulting

(15) |

A Proportional Derivative (PD) type multivariable controller can now be used, that is

(16) |

where is the desired set point for the vector output ** y**(

*t*) and is the controller gain matrix with (computation of matrix

**will be considered in section II.C). The estimation error in the ESO is computed as**

*K*(17) |

Using (11) and (12) the estimation error of the ESO is

(18) |

By replacing (16) in (13), we can express as

(19) |

And then combining (10), (18) and (19) yields the closed-loop MIMO-ADRC equations

(20) |

From (20), it is straightforward to verify that the eigenvalues of the system matrix of the closed-loop MIMO-ADRC equations are given by the eigenvalues of and (** A-LC**). Since it can be shown that the pair is controllable and the pair (

**) is observable, the stability of (20) can always be ensured by placing the controller and observer poles appropriately. Moreover, under the assumption of boundedness of , the BIBO stability of (20) is assured (Zheng**

*A,C**et al*. 2007). This is the case when or its rate of change is small.

**B. LQR formulation of the MIMO-ADRC problem **

In order to develop the LQR formulation of the MIMO-ADRC problem, we write the control law in (16) in terms of the real state vector

(20) |

that is

(21) |

It must be remembered that because of the separation principle we can always apply (16) with the estimate of the state to the real system. Defining the vector

(22) |

with the desired set point, we can establish the tracking error as

(23) |

Taking the derivative of (23) and using (15) yields

(24) |

By adding and subtracting the term in (24) we have

(25) |

It can be easily shown that , then the tracking error is computed as

(26) |

In order to have the LQR formulation of the ADRC problem given by (26) we define the quadratic cost

(27) |

where ** Q **and

**are respectively; a positive semi-definite and a positive definite matrices. It is well known that the minimization of (27) gives the state-feedback control**

*R*(28) |

That is, we have obtained the original control law (21) via the LQR approach. Here, the feedback matrix is

(29) |

where ** P **is the symmetric positive definite solution of the Continuous Algebraic Riccati Equation (CARE) given by

(30) |

In Fig. 1, it is shown the MIMO-ADRC configuration.

Figure 1. MIMO-ADRC tuning employing LQR

**C. Selection of matrices Q and R **

The matrices and in (27) are the tuning parameters for computing the matrix ** K **in (29). In LQR, the connection to closed-loop dynamics is indirect; it depends on the choice of matrices

**and**

*Q***, this implies it is usually necessary to perform some trial and error before to obtain a satisfactory closed-loop response. For this reason, it is interesting to link LQR to pole placement by requiring that the closed-loop poles of the MIMO-ADRC system (26) with optimal control law (28) lies in some specific region of the complex plane. A simple and very useful example of this, is when we require that the closed-loop poles have real part to the left of**

*R**s=-α*for

*α*∈ℜ

^{+}. In LQR theory, this problem is called LQR design with a prescribed degree of stability (Anderson and Moore, 1989). In the following result, it is adapted to MIMO - ADRC.

**Theorem 1: **Let a MIMO-ADRC system be described by the state equations (26) and the LQR criterion

(31) |

then the eigenvalues of the closed-loop matrix lie in ℜ(*s*)<-*w _{c}*, where

*w*>0 and the control signal is

_{c}

*u*_{0}=-

*Kx**with gain matrix nd*

**the symmetric positive definite solution of the CARE given by**

*M*(32) |

**Proof: **Replacing the coordinate transformation and in (31) gives the standard LQR criterion (27).

Algorithm 1 shows how to implement theorem 1 through the Matlab_{®} command *lqr*.

**Algorithm 1: **Matlab LQR-MIMO-ADRC design for a prescribed degree of stability *w _{c}*

*Input*: from (10). ** Q**,

**,**

*R**w*(31).

_{c }from*Step 1*:

*Step 2*:

*Output*:

Another typical region is when we require the closed-loop poles to be inside a circle with radius *ρ* and with center at (-*α*,0) with *α*>*ρ*≥0. That is, the circle *C*(*α*,*ρ*) is entirely within the left-half plane. This can be achieved by first transforming the Laplace variable *s *to a new variable *σ*, defined as *σ=*(*s+α*)/*ρ*. This takes the original circle in *s*-plane to a unit circle in *σ*-plane. The corresponding, transformed state-space model has the form

(33) |

One then treats (33) as the state-space description of a discrete-time system. So, solving the corresponding discrete optimal control problem leads to a feedback matrix such that has all its eigenvalues inside the unit disk. This in turns implies that, when the same control law is applied in continuous-time, then the closed-loop poles reside in the original circle in *s*-plane. The above result is summarized in Algorithm 2.

**Algorithm 2: **Matlab LQR-MIMO-ADRC design such that the closed-loop poles are inside the circle *C*(*a*,*r*)

*Input*: from (10). ** Q**,

*R**from*(27).

*Parameters α*,

*ρ of the circle.*

*Step 1*:

*Step 2*:

*Step 3*:

*Output*:

**Remark 1: **The above ideas can be extended to other cases, in which the desired region can be transformed into the stability region, see for example (Misra, 1996).

**D. LQR-ESO design**

ADRC method is based on the separation principle, this allows us to treat the unknown dynamic and disturbances in a physical process as the generalized disturbance, built an ESO to estimate it in real-time, and then canceling its effect using the estimate as part of the control signal. For computing the ESO gain matrix with *L** _{i}*∈ℜ

*in (12) within an LQR formulation, we propose the LQR design with a prescribed degree of stability (Anderson*

^{pxp}*et al*., 1989), this allows us to stay in the LQR framework and impose the practical condition that the ESO dynamics must be faster than the controller one. By duality, it is simple to adapt theorem 1 to design an optimal ESO with a prescribed degree of stability by replacing

**←**

*A*

*A**,*

^{T}**←**

*B*

*C**and*

^{T}**←**

*K*

*L**.*

^{T}**Theorem 2: **Let an ADRC system where the control law *u*_{0}(*t*)* *in (28) has been computed such that the closed-loop poles are inside the region ℜ(*s*)<-*w _{c}* and the LQR with a prescribed degree of stability criterion is

(34) |

Then the ESO described as (12) where *w*_{0} is the required ESO bandwidth chosen as *w*_{0}=*γw _{c}* has a gain matrix given by

(35) |

that places the ESO eigenvalues into *ℜ*(*s*)<-*w*_{0} with ** M **the symmetric positive definite solution of the CARE given by

(36) |

**Remark 2: ***w*_{0} is chosen *γ *times the maximal possible closed-loop pole that is defined through *w _{c}* in Theorem 1.

For ease of reference we summarize the procedure of design in Algorithm 3.

**Algorithm 3: **Matlab LQR-ESO design with a prescribed degree of stability

*Initialization: **Q*_{0}=*I*_{p}_{(n+1)}, *R*_{0}=*I*_{m}

*Input*: *A,C**from *(12). *Q*_{0},* R*_{0}, *w*_{0} *from *(34).

*Step 1*:

*Step 2*:

*Step 3*:

*Output*:

Under the assumption of controllability of the pair in (10) and by a suitable choice of the feedback matrix ** K **in (16) (Algorithms 1 or 2), the closed-loop poles could be assigned to any desired set of locations. However, if the closed-loop poles are chosen much faster than those of the plant, then the gain

**will be large, leading to a large plant input. A similar problem arises in the ESO design (Algorithm 3). If we consider the presence of measurement noise**

*K***(**

*η**t*) in the controlled output

**(**

*y**t*), the estimation error dynamics of the ESO (18) takes the form

(37) |

It is evident that a large value for ** L **will enhance the effect of the measurement noise, since this is usually a high-frequency signal. We then need a compromise between speed of response and noise immunity. In MIMO-ADRC, for controller design, we place the desired closed-loop poles in the region

*ℜ*(

*s*)<-

*w*based on typical performance criteria (rise time, settling time, overshoot, etc.). The larger the parameter

_{c}*w*, the faster the response, the larger the control signal and a system more susceptible to noise. For ESO design, we choose the ESO bandwidth

_{c}*w*

_{0}by fixing the factor

*γ*in

*w*

_{0}=

*γw*to be two to six times faster than the controller poles. This ensures the observer errors decay faster than the desired closed-loop dynamics allowing the controller poles to dominate the total response. If sensor noise if a problem or there are actuator constraints then the observer poles may be chosen slower than two times

_{c}*w*. This would yield a system with lower bandwidth, more noise smoothing and less control energy expenditure.

_{c}**III. NUMERICAL EXAMPLE**

Consider a system (Goodwin *et al., *2000) having the transfer function matrix.

(38) |

The constants *k*_{12} and *k*_{21 }depend on the operating point, a common situation in practice. We will consider in Table 1, four operating points, where it is computed the relative gain array (RGA) (Bristol, 1966) to measure the interaction between the loops. We will use in all the cases the pairing (*u*_{1},*y*_{1}), (*u*_{2},*y*_{2}).

**Table 1**: Operating points for system (38)

The system (38) can be represented in the form (6) employing the inverse Laplace transform and superposition as

(39) | |

(40) |

For MIMO-ADRC design, we write (39) and (40) in the form (7) selecting and . The state space description for vector is

(41) |

Where we have done to remove the scaling matrix. The extended state space description of (41) for is

(42) | |

The ESO is obtained through the equations

(43) |

where is the ESO gain matrix which is computed through algorithm 3 for *Q*_{0}=*I*_{6}, *R*_{0}=*I*_{2} and *w*_{0}=5. The control input is

(44) |

where is the controller gain matrix and *z*_{3}(*t*) is the estimate of the generalized perturbation . The gain matrix ** K **is computed employing the algorithm 1 by choosing

**=**

*Q*

*I*_{4},

**=**

*R*

*I*_{2}and

*w*=1.

_{c}Figure 3 shows the set point tracking performance of the proposed method considering the four operating points described in Table 1. In all the cases we apply a set point in channel 1 with amplitude 1 at *t*=1 and a set point in channel 2 with amplitude -1 at *t=*10. The situation where a change in the reference in one loop has an effect more significant in the output of the other loop corresponds to the strong interaction case but in general, we can say that the results are very acceptable even for the case of strong interaction. Moreover, the RGA for the situation of strong interaction indicates that we should change to the pairing (*u*_{1},*y*_{2}), (*u*_{2},*y*_{1}). However, MIMO-ADRC allows using non-dominant control signals and this is not necessarily detrimental.

Figure 3. Set-point tracking using MIMO-ADRC for the four operating points in Table 1.

We now compare MIMO-ADRC design with the standard decentralized control (Goodwin *et al., *2000; Albertos and Sala*, *2004; Skogestad and Postlethwaite, 2005). To be specific, we aim that the two loops have the same complementary sensitivity given by the transfer function 9/(*s*^{2}+4*s*+9). The corresponding controllers are *C*_{1}(*s*)=(4.5*s*^{2}+13.5*s*+9)/(*s*(*s*+4)) and *C*_{2}(*s*)= (1.5*s*^{2}+ 7.5*s*+ 9)/ (*s*(*s*+4)). Figure 4 shows the performance of both methods for the same tracking conditions considered previously and for the case of weak interaction between the loops. We see that the interaction has an effect more important in decentralized control and this is even more dramatic in the case of strong interaction as it is shown in Fig. 5.

Figure 4. Set-point tracking for MIMO-ADRC and decentralized control for the case of weak interaction.

Figure 5. Set-point tracking for MIMO-ADRC and decentralized control for the case of strong interaction.

The standard decentralized control becomes unacceptable for some operating points of the example. On the other hand, in MIMO-ADRC design a single controller can cover all the operating points without any post tuning of the controller's parameters.

**IV. CONCLUSIONS **

In this paper, a LQR solution has been used to develop optimal tuning algorithms for decoupling uncertain MIMO systems that have been formulated into the ADRC framework. The desired performance objectives are globally and directly addressed by minimizing a quadratic function of the state and control input avoiding the typical trial-and-error strategy of SISO-ADRC for every loop of the MIMO system.

**ACKNOWLEDGMENTS **

The first author would like to thank the *Laboratoire d'Analyse et d'Architecture des Systèmes *(LAAS - CNRS) for hosting him during the development of this work.

**REFERENCES**

1. Albertos, P. and A. Sala, *Multivariable Control Systems: An Engineering Approach*, Springer (2004). [ Links ]

2. Anderson, B.D.O. and J.B. Moore, *Optimal Control: Linear Quadratic Methods*, Prentice-Hall (1989). [ Links ]

3. Bristol, E., "On a new measure of interaction for multivariable process control," *IEEE Trans. Auto. Control*, **11**, 133-134, (1966). [ Links ]

4. Falb, P. and W. Wolovich, "Decoupling in the design and synthesis of multivariable control systems," *IEEE Trans. Auto. Control*, **12**, 651-659 (1967). [ Links ]

5. Friedly, J., "Use of the Bristol array in designing noninteracting control loop: a limitation and extension," *Ind. Eng. Chem. Process Des. Dev.*, **23**, 469-472, (1984). [ Links ]

6. Gao, Z., Y. Huang and J. Han, "An Alternative Paradigm for Control System," *IEEE Conference on Decision and Control*, **5**, 4578-4585, Orlando, USA (2001). [ Links ]

7. Gao, Z., "Scaling and Bandwidth Parameterization Based Controller Tuning," *American Control Conference*, 4989-4996, Denver, CO, USA (2003). [ Links ]

8. Goodwin, G., S. Graebe and M. Salgado, *Control System Design*. Prentice-Hall (2000). [ Links ]

9. Han, J., "Auto-Disturbance Rejection Control and its Applications," *Control and Decision* (in Chinese), **13**, 19-23, (1998). [ Links ].

10. Han, J., "Nonlinear Design Methods for Control Systems," *14th IFAC World Congress*, Beijing, China (1999). [ Links ]

11. Han, J., "From PID to active disturbance rejection control," *IEEE Trans. on Indust. Electronics*, **56**, 900-906, (2009). [ Links ]

12. Huang, H., L. Wu, J. Han, G. Feng and Y. Lin, "A new synthesis method for unit coordinated control systems in thermal power plant - ADRC control schema,"* International Conference on Power System Technology*, Singapore (2004). [ Links ]

13. Khani, F. and A. Yazdizadeh, "Boiler-Turbine unit controller design based on the extended state observer," *IEEE International Conference on Control and Automation*, New Zealand (2009). [ Links ]

14. Kwakernaak, H. and R. Sivan, *Linear Optimal Control Systems*, Wiley Interscience (1972). [ Links ]

15. Lau, H., J. Alvarez and K. Jensen, "Design of control structures by singular value analysis," *AIChE J.*, **31**, 70-75 (1989). [ Links ]

16. Liu, B., Y. Xia and J. Chen, "Decoupling double-loop control system with ADRC," *3rd ^{ }International Conference of Innovative Computing Information and Control*, Kaohsiung, Taiwan (2008). [ Links ]

17. Misra, P., "LQR design with prescribed damping and degree of stability,"

*IEEE International Symposium on Computer-Aided Control System Design,*Michigan, USA (1996). [ Links ]

18. Miklosevic, R. and Z. Gao, "A dynamic decoupling method for controlling high performance turbofan engines,"

*16th IFAC World Congress*, Czech Republic (2005). [ Links ]

19. Niederlinski, A., "A heuristic approach to the design of linear multivariable interacting control systems,"

*Automatica*,

**7**, 691-701 (1981). [ Links ]

20. Rosenbrock, H., "Design of multivariable control systems using the inverse Nyquist array,"

*Proc. IEEE*,

**16**, 1929-1936 (1969). [ Links ]

21. Shi, R., J. Song and W. Wan, "Dynamic modeling and active disturbance rejection controller design for a morphing wing mechanism,"

*International Conference on Modeling, Identification and Control*, Wuhan, China (2012). [ Links ]

22. Shinskey, F.,

*Process Control Systems*, McGraw-Hill Book Company (1979). [ Links ]

23. Skogestad, S. and I. Postlethwaite,

*Multivariable feedback control: Analysis and Design*, Wiley (2005). [ Links ]

24. Tian, L., D. Li and C. Huang, "Decentralized controller design based on 3-order active-disturbance rejection control,"

*World Congress on Intelligent Control and Automation*, Beijing, China (2012). [ Links ]

25. Xia, Y., P. Shi, G. P; Liu, D. Rees and J. Han, "Active disturbance rejection control for uncertain multivariable systems with time-delay,"

*IEEE Control Theory Appl.*,

**1**, 75-81 (2007). [ Links ]

26. Zheng, Q., L. Gao and Z. Gao, "On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknown dynamics,"

*IEEE Conf. Decision Control*, 3501-3506 (2007). [ Links ]

27. Zheng, Q., Z. Chen and Z. Gao, "A practical approach to disturbance decoupling control,"

*Control Engineering Practice,*

**17**, 1016-1025 (2009). [ Links ]

**Received: March 24, 2013 Accepted: May 27, 2014 Recommended by Subject Editor: José Guivant**