The design method of simple adaptive control system with PFC for MIMO system via DE

ABSTRACT This paper deals with a design method of adaptive output feedback control system with parallel feedforward compensator (PFC) for a multiple input multiple output system. An adaptive output feedback control system, which is based on almost strictly positive real (ASPR), requires PFC in most cases. The design method is using one-shot input/output data of the controlled system, and parameter optimization will be done by differential evolution (DE). By using DE, the designer can optimize the design parameters without trial and error, which can be a more practical method.


Introduction
Adaptive control systems based on almost strictly positive real, typified by simple adaptive control (SAC) [1], are well known for high robustness with respect to uncertainness and disturbances [1,2].On the other hand, there has been a problem that ASPR conditions are severe for an actual system.In order to solve this problem, the introduction of parallel feedforward compensator (PFC) has been proposed [3].It is possible to configure adaptive control based on ASPR by PFC that makes the augmented system ASPR.Several design methods of such PFC have been proposed and widen a scope of application of ASPR-based adaptive control [3][4][5][6].In addition, the methods of designing PFC from the input/output data of controlled system have recently been proposed [7][8][9][10].Thus, ASPR-based adaptive control systems became more practical control method.However, another problem is how to determine the parameters of adaptive adjustment law.If the control system is ASPR, stable output can be obtained, but the control performance in the transient state depends on the parameters of the adaptive adjustment law.Therefore, the parameters of the adaptive adjustment law must be appropriately determined.These parameters are usually determined by trial and error through numerical simulations.Therefore, determining appropriate parameters is difficult and requires enough experience.This paper proposes the PFC design method and the parameter determination method for the adaptive adjustment law of SAC.The PFC will be designed to output the ideal PFC output derived from the oneshot input/output data of controlled system.In other words, ideal ASPR model, filter order and parameters which are needed in PFC design are optimized so as to minimize an evaluation function that compares ideal PFC output and designed PFC output.The PFC design method is proposed in SICE Annual Conference paper [10] and it is shown the usefulness through numerical simulations.In this paper, the method which designs not only PFC but also whole SAC from the one-shot input/output data will be proposed.By the method, the designer could be able to design a SAC without prior experience.Specifically, a nominal model is derived from the designed PFC and the ideal ASPR model, and the parameters of the adaptive adjustment law are optimized to minimize the output error.In the proposed method, for the parameters optimization, differential evolution (DE) [11] is adopted.DE is reported to outperform genetic algorithms, particle swarm optimization [12][13][14], and other techniques in convergence speed and robustness [15].For this reason, some research adopted DE to optimization of control system [16,17].Covariance matrix adaptation evolution strategy ( CMAES) is also used in data-driven design [18].However, DE has much simpler algorithm than CMAES and easy to use [19].From the above reasons, DE is adopted in this paper.

Problem statements
Consider the following nth order multiple input multiple output (MIMO) stable system: where x(t) ∈ R n is a state vector, u(t), y(t) ∈ R m are the input and output of the system, respectively.The transfer function representation of the above system is given as follows: For this transfer function, the following PFC which is parameterized by ρ is added.
Here, the state-space representation of the PFC is as follows: where x f (t) ∈ R n f is a state vector, u f (t), y f (t) ∈ R m are the input and output of the system, respectively.Then the augmented system G a (s) with PFC could be expressed as follows: ( For this augmented system, consider adopting the following SAC with Integral + proportional adaptive adjustment law [1].The block diagram is shown in Figure 1. ) K e (t) = K eI (t) + K eP , ( 13) where is the augmented system output error with the reference signal y m (t) and the reference signal is generated by the following model: Also, the parameters of the adaptive adjustment law are Here, note that, H(ρ) = O (zero matrix) when ρ = 0.
With the above setting, the objective is to design PFC from the input/output data u 0 (t) and y 0 (t), so that the augmented system with PFC is ASPR.Note that the input/output data can be either open-loop or closedloop, since the controlled system is stable.To achieve this objective, define the ideal ASPR model G * a and ASPR output as follows: Now consider obtaining the parameter ρ which is minimizing the difference between augmented system output y a ∈ R m and ASPR model output y * a ∈ R m .Therefore, consider minimizing the following evaluation function: Furthermore, using the PFC that minimizes the above evaluation function, consider adaptive adjustment law parameters that minimize the following evaluation function:

Specific design of PFC [10]
Define the ideal PFC output y * f ∈ R m as follows: Then, assume that ideal PFC H * (s) is specified as follows: From this transfer function, the ith line output can be expressed as follows: Then by introducing the following n h th order stable filter, the ideal PFC output can be obtained as follows: Here, and However, the ideal parameter ρ * is unknown, that is, ρ * i is unknown.Therefore, with the parameter ρ, define the PFC output as follows: where Then finally consider minimizing the following evaluation function: This can also be expressed as follows: Here, the following equation holds: Thus, the parameter ρ which minimizes the evaluation function (33) also minimizes the evaluation function (20).The minimization problem of evaluation function depends on the order of designed PFC and the stable filter equation (26) order and parameter.Also, the ideal ASPR system model effects to evaluation function.These parameters are conventionally designed by trial and error by the designer.However, that requires experience and knowledge and is sometimes quite difficult even if you have enough experience.Therefore, the parameter optimization by DE will be considered in the next section.

Determination of parameters for adaptive adjustment law
To optimize the parameters of adaptive adjustment law, some kind of nominal model is needed.From Equation ( 5), the nominal model G(s) can be obtained as follows: By using this nominal model, consider minimizing the evaluation function (21).Here, the parameters e , xm , xu , which minimize the evaluation function (21), will be determined by DE.

PFC design via DE
In this section, we discuss about the parameter optimization for PFC design via DE.Although DE is a simple algorithm, it is reported to outperform in convergence speed and robustness [15].
The parameter of filter equation (26), PFC equation (32) and ASPR system model will be optimized.For simplicity, ASPR system models are defined as following first-order transfer function; Here, I ∈ R m×m is the identity matrix.Thus, the parameters of the ASPR system model are K and T. All of the parameters are needed to be positive value.Therefore, the sigmoid function is applied.
The parameter X will be searched in DE.Then the parameters of filter, PFC and ASPR model will be searched in the range of [S min S max ].DE algorithm adopted in this paper is explained below.

• Step 1: Generation of initial individuals
As a target vector, a vector of the same dimension as the number of parameters is generated by a uniform random number within a predetermined range.For example, X (1)  k = X (1) = ρ (1)   1 • • • ρ (1)  n h K (1) T (1) , where D is the dimension of decision variable vector and superscript g of X (g) k is the generation number.Here, as the number of variables to be searched increases, the required number also increases, so the number N depends on the order n h .

• Step 2: Extraction of base vector
Extract the best vector from gth generation vector which minimizes the evaluation function J.This is, set the base vector as follows when the lth target vector minimizes the evaluation function. (40) • Step 3: Mutation Generate a mutated vector from randomly chosen two vectors as follows; where r1, k and r2, k are generated by using uniform random numbers and F is a scaling factor for scaling the difference vector.
• Step 4: Crossover Generate a mutated vector as follows: where C ⊆ [0, 1] is crossover rate, rand k,o is an uniform random number within [0, 1] and o r is a random value within [1, D].Then generate trial vector as follows: • Step 5: Survivor selection Select the survivor vector for next generation as follows: (44)

• Step 6: Iteration
Steps 2-5 are iterated until k = L and the best vector X (L) best of Lth generation is adopted as final solution.
• Step 7: Update of PFC order Update the order of PFC n h to n h + 1 and do Steps 1-6.If the value of evaluation function for best vector X (L)  best of PFC order n h + 1 is larger than n h , the best vector X (L) best obtained at the order n h is adopted.

Determination of parameters for adaptive adjustment law via DE
After designing the PFC, the numerical simulations will be done to determine the parameters of SAC by using the nominal model (36).Here, the parameters for adaptive adjustment law will be optimized via DE so that the evaluation function ( 21) is minimized.As mentioned, the parameters I and P of SAC have to be positive definite symmetric matrixes.Therefore, in the same way of PFC, the sigmoid function ( 38) is applied to search positive values.Then Steps 1-6 of DE algorithm which are mentioned above will be done.

Numerical simulations
In this section, numerical simulations are performed to confirm the effectiveness of the proposed method.The simulations are done by MATLAB/Simulink.The following two-input/two-output system is considered: where transfer function representation is given as follows: Note that the system is basically unknown.Here, one of the required condition is CB > 0. It can be seen that the controlled system does not satisfy this condition.
Input/output data may be either open-loop or closed-loop, but the design here is based on the assumption that closed-loop data are obtained.To obtain input/output data u 0 = [u 01 u 02 ] , y 0 = [y 01 y 02 ] of this system, following output feedback controller is designed: The obtained input/output data are shown in Figure 2. From these data, PFC will be designed.
For designing a PFC, the ideal ASPR model is set as Equation (37).The ideal ASPR model's gain K and time constant T will be searched for by DE.In many cases, the setting of scaling factor F ≥ 0.6 and crossover rate C ≥ 0.6 leads to results having better performance [20].Here, the scaling factor and crossover rate are set to the same values as in Refs.[9,10].DE parameters are set as scaling factor F = 0.8, crossover rate C = 0.9 and number of generations L = 300.The initial population was generated five times as big as D. Then the parameters of filter, PFC, ASPR model will be obtained using Equation (38) as follows: Figure 3 shows the evolution of the value of the evaluation function.It can be seen that the value of evaluation function J converges to 0 around the 70th generation.Here, the value of evaluation function was greater at the order n h = 2 as compared to n h = 1, and optimization was terminated.
By using the first generation parameters, the outputs of ideal ASPR model, ideal PFC, actual augmented system and actual PFC which uses the input shown in Figure 2 are obtained as Figure 4. Also, the outputs by the 300th generation parameters are shown in Figure 5. From Figure 5, it can be seen that the output of ideal ASPR and the actual augmented system is exactly matched.
By the optimized parameters, which is the 300th generation parameters, ideal ASPR model and PFC were obtained as follows: Then, the state-space representation of the PFC is given as follows:  By using the state x a (t) = [x(t) x f (t) ] , the augmented system is defined as follows: (52) This system satisfies the condition of C a B a > 0 and invariant zeros were −2.2583 ± i1.3675 and −2.6167 ± i0.0381, that is, stable zeros.Note that it is usually unknown.
For the system with the designed PFC added, the SAC adaptive adjustment law was determined via DE.Here, for simplicity, the adaptive control law is set as follows: The parameters of adaptive adjustment law will be obtained by using Equation (38) as follows:

Conclusions
In this paper, one of the design method of the adaptive control system with PFC for MIMO system was proposed.Also, the optimization method of design parameters was shown via DE.The effectiveness of the proposed method was confirmed through numerical simulations.It showed that the proposed method could decide appropriate parameters for ASPR model, PFC and adaptive adjustment law.Also, the parameters of the adaptive adjustment law were optimized using the nominal model derived via the PFC designed from the input/output data.In order to make the method high generalization performance, it is necessary to judge whether the obtained PFC can be converted to ASPR.That is the research task from now on.

Figure 1 .
Figure 1.Block diagram of the control system.

Figure 3 .
Figure 3.The values of the evaluation function of first-order PFC and second-order PFC. b

Figure 4 .
Figure 4. (a) Outputs y 1 (t) first generation parameters.(b) Outputs y 2 (t) by first generation parameters.The output data by first generation parameters.

Figure 7 .
Figure 7. (a) Outputs y 1 (t) by generation parameters.(b) Outputs y 2 (t) by 300th generation parameters.The control results by the adaptive control system.