Modal synthesis of precision control systems

. The problem of synthesis of precision modal control systems is considered. It is noted that a common approach to solving this problem is to consistently meet the requirements for the nature of the transient process and for the indicators of its accuracy. This approach to synthesis is faced with the need to make design decisions under incomplete conditions. In practice, this circumstance leads to obtaining synthesis results with undesirable deviations from technical requirements. When designing precision control systems, such deviations are unacceptable. To eliminate the difficulties that arise, a transition to interval methods for formulating and solving modal synthesis problems is proposed. The theoretical possibility of the interval approach is based on the excessive variety of possible placement of eigenvalues in the spectrum of the characteristic matrix of the system. An example of an interval synthesis of a system with a modal controller and additional output feedback is considered. The restrictions on the spectrum of the specified matrix are formed, which determine the fulfillment of the requirements for the monotonicity of the transient process, the regulation time and the accuracy of the response to harmonic influences. It is noted that the variety of solutions obtained creates the preconditions for a multi-alternative approach to modal synthesis of systems .

The problem of forming the spectrum of eigenvalues of the characteristic matrix arising during the synthesis of such systems is traditionally solved without taking into account the requirements for the accuracy indicators of the response to the reference input signals. These indicators are ensured at the subsequent stages of system design by introducing an external signal model and additional correcting filters into its structure [11][12][13][14][15][16][17][18].
Such a sequential approach to the synthesis of a control system is based on the widespread method of decomposition of the general design problem, and is convenient in methodological terms. However, the separate provision of the requirements for the time indicators of the transient process and the indicators of the system accuracy leads to their incompatibility. The most frequent manifestation of this incompatibility is the occurrence of unacceptable oscillability of the transient process during stepwise effects on the system. Eliminating oscillability due to the deterioration of the accuracy of the system comes down to finding a compromise. It is undesirable in the general case and impossible when synthesizing high-accuracy systems.
In this paper, a method for the synthesis of modal control systems is presented, which guarantees the joint provision of the required quality of the transient process and the accuracy of the system. Synthesis procedures will be considered using two typical structures as an example:  structure with output feedback;  structure with an additional input channel for the reference input signal.
2 Synthesis of control in a structure with output feedback

Synthesis method
Let a completely controllable and, in the general case, unstable object be known: for which the following are defined: B -characteristic matrix, N -control matrix, Aoutput matrix, x -state coordinate vector, u -control, y -controlled variable.
For this object, we set the task of synthesizing a modal control system with the following quality indicators:  monotonous nature of the transition process;  regulation time tr;  zero steady-state error for the constant reference input signal g(t) = const, i.e. first order astatism with respect to the input signal;  the maximum allowable error v for the velocity action g(t) = vmaxt, i.e. quality factor Dv=vmax/v in speed;  maximum permissible relative errors δ1 and δ2 of reproduction of the amplitude of harmonic influences at frequencies To solve this problem, a system is formed with the structure shown in figure 1. Where: R = [r1 r2 …rn]modal regulator; d and k/sadditional inertialess and integrating links introduced into the system to ensure the above quality indicators [19,20].
For the structure in figure 1, let us find the equation of motion in the Laplace images: The characteristic polynomial of the system is obtained from (2): The transfer function of the closed system takes the form: from which it directly follows that the system has the property of astatism, i.e. already at the structural level, for any parameters of the system, a zero steady-state error is ensured for constant actions g(t) = const [21].
Since the roots of polynomial (3) are functions of free parameters of the system s1(r1,…,rn, k,d), …,sn+1(r1,…,rn, k, d), and the number of variables (n+2) is greater than number of roots (n+1), then it becomes possible not only to place the roots of the polynomial in the desired way (n+1 equation), but also to provide the required quality factor Dv in speed: A system of n+2 equations with n+2 unknowns will be obtained. In this case, the relative errors δ1 and δ2 of the reproduction of harmonic influences remain undefined.
To increase the number of free variables, as well as reduce the computational complexity of solving the resulting multidimensional problem, we use two methods:  we pass from equations to interval conditions, i.e. compose a system of inequalities that reflect the requirements for the accuracy of the system and the nature of the transient process;  we restrict ourselves to two-sided interval specification of only a part of the roots of the characteristic polynomial that have a dominant value and make the remaining roots insignificant, moving them away from the dominant ones using one-sided inequalities.
As a result, we get the following problem: where: Ωthe desired geometric mean root determined by the given control time; coefficient that determines the length of the interval;  -coefficient specifying the degree of removal of insignificant roots from dominant ones,  > ; W(ωz,g)the modulus of the frequency function W(jω) at the frequency ωz,g; zthe ordinal number of the harmonic influence.
Thus, the solution of the problem of ensuring the accuracy of harmonic and monotonic control actions with the simultaneous fulfillment of the requirements for the nature of the transient process and the regulation time was reduced to the fulfillment of a system of restrictions (6).

Method verification and discussion of results
Let's consider an example of solving problem (6). Unstable object is set: Let us synthesize a system with a monotonic character of the transient process and a regulation time tr=0.05 s.
For g(t)=constit is necessary to provide first-order astatism with a speed quality factor Dv 30 s -1 .
The harmonic signal g(t)=0.5sin(3t) needs to be reproduced with a static error δ1  0.01, and the signal g(t)=0.5sin(30t)with a static error δ2  0.1.
For two dominant roots and the regulation time tr, we define the geometric mean root Ω: where: Tconstant depending on the order of the desired polynomial. Let's take  = 100 s -1 and choose  = 1.2;  = 5.
For the given conditions, solution (6)   Transient processes in the system are shown in figure 2 and figure 3.  The requirements for the quality indicators of the transient process and for the accuracy of the response to monotonic and harmonic influences are met. It should be noted that the use of interval conditions of the synthesis problem makes it possible to obtain several feasible solutions depending on the initial point of the search. This circumstance determines the possibility of the subsequent choice of one of the obtained solutions according to additional criteria that are not taken into account in the formal setting of the problem. In the general case, this possibility allows proceeding to the formulation of a multi-alternative synthesis problem [22][23][24][25][26].

Synthesis method
Let's move on to considering a more perfect structure of the control system, which allows providing high-order astatism in it, i.e. zero error not only for constant, but also for velocity actions of the form g(t) = g0 + vt, figure 4.
In this structure, the partial invariance of the system error (up to the second derivative of the reference input signal g(t)) is realized by choosing the corresponding values of the parameters с0 and с1 of the additional reference input channel. To analyze the dynamic properties of the system shown in figure 4, let's consider an object with a differential equation containing the derivatives of the control action on the right side: where a0,…, an-1, b0,…,bmconstant coefficients. As a result of constructing a modal controller R = [r1 r2 …rn], the equation of motion of the closed part of the system will take the form: in which the coefficients a0,M,…, an-1,M determine the desired location of n roots of the characteristic polynomial -the poles of the system. In this case, the polynomial of the numerator of the transfer function of the closed-loop system contains roots (zeros of the system) that can dominate over its poles. In the general case, we can assume that some part of the poles (set S1) forms the required character of the transient process and regulation time, and the remaining |S2| free roots (|S2| = n-|S1|) can be located at a sufficient distance from S1 so that the roots of S2 do not significantly affect the quality of the transient process.
We now introduce the observer of the first derivative of the reference input signal at the input of the system, see figure 4.
For the resulting complete structure, a factor c1s+c0 appears on the right-hand side of equation (10), which coefficients с0 and с1 (in the case of compensation of the object's zeros) must satisfy the condition of the second-order astatism c1s+c0 = a1, M s+a0, M.
The transfer function of the system will take the form: If the zero ( = -c0/c1) of the transfer function W(s) turns out to be dominant with respect to the poles S1 of the closed-loop system, then during the response to the stepwise input action, oscillability (overshoot) of the output value y(t) will occur.
To eliminate this oscillability, it is necessary to compensate for zero (-c0/c1), i.e. introduce an additional pole s2  S2 corresponding to the condition s2  (-с0/с1) into the desired polynomial of the system.
In this case, the uncertainty of the system synthesis problem arises: to build a modal controller, it is necessary to know the parameters of the reference channel с0 and с1, and to build the reference channel, it is necessary to know the characteristic polynomial of the in such a way that the inequality   (-а0,M/а1, M) holds [27][28][29]. Restricting ourselves to the case of negative real roots that is important for this problem, we prove the following theorem. Let denote  through an arbitrary root S. Then, to prove the theorem, it is sufficient to establish the inequality: Since all roots are negative, inequality (14) holds for all roots of the polynomial under consideration, i.e. the value (-а0/а1) is their upper limit.
The theorem is proved.
Corollary. The theoretical inaccessibility of the condition of full compensation   (-а0,M/а1,M), or, equivalently, s2  (-с0/с1), nevertheless leaves the practical possibility of an arbitrarily close approximation to its fulfillment. This possibility is indicated by inequality (14), from which it follows that choosing the root s2 =  dominant (maximum) in the desired characteristic polynomial and placing all other roots si s2, (i = 1,..,n-1) in accordance with the condition si << s2, one can come to the relation: where the quantity  determined by the condition si << s2 can be chosen arbitrarily small. The tendency of inequality (15) to equality will mean the approach to the fulfillment of the compensation condition s2 = (-а0,M/а1,M).
Thus, on the basis of the above interval analysis, we can propose the following synthesis technique for a modal control system, performed according to the structure in figure 4: 1) one or a group of multiple roots S1 is determined based on the specified control times (see (8)); 2) the root s2 with a dominance coefficient, for example,  > 100, is determined with respect to the roots S1; 3) the remaining free roots are made insignificant by placing them at a sufficient distance to the left of the roots S1 and S2; 4) a modal regulator, which forms the desired characteristic polynomial which eliminates the influence on the characteristics of the system of zeros of the object's transfer function.

Method verification and discussion of results
Let us demonstrate the efficiency of the method using a numerical example.
Consider an object given in the form (1)  The numerator of the transfer function of the object is 17784000(s+44), i.e. its transfer function contains zero  = -44. The roots of the object are in the interval (-24944; -0.036).
The system requires a monotonic transient process with a regulation time tr = 0.01 s and second-order astatism.
From the given tr we determine s1: Let's determine the compensating root s2 from the condition s2  100s1: s2 = -1.
Analysis of figure 5 shows that, despite the absence of complex roots of the characteristic polynomial, a significant overshoot  = 30% occurs in the system without compensation of the observer's zero. Application of the proposed synthesis method practically eliminates oscillability, reducing overshoot to  = 0.4%.  The presence of second-order astatism in the system is confirmed by the constant value of the steady-state error (t) = g(t)y(t) when the input signal is supplied with constant acceleration: g(t) = t 2 , figure 6.
It follows from figure 6 that the quality factor of the acceleration system is: Thus, in a system with modal control, the joint fulfillment of the requirements both to the nature of the transient process and to the indicators of its accuracy is ensured without the use of integrators in a closed loop.

Conclusion
The problem of ensuring the accuracy of modal control systems is traditionally considered as an independent stage of system design, not associated with the stage of calculating the modal controller itself.
The research results presented in the paper show that the indicated approach to modal synthesis of systems leads to problems with incomplete conditions: the solution of each of them requires a preliminary solution to the other.
Ignoring the noted interdependence greatly simplifies the practical synthesis of systems, but at the same time, it becomes necessary to find a compromise between the requirements for the nature of the transient process and the requirements for accuracy indicators. Such a compromise leads to undesirable, and in the case of designing systems with a high astatism order -to unacceptable deviations from the technical specifications.
The proposed method for designing modal control systems eliminates the indicated uncertainty of synthesis subproblems, solving them as a single multidimensional problem of ensuring interrelated quality indicators of the system.
Practical methods of such an approach to synthesis, demonstrated using examples of typical structures with additional output feedback and with a differentiating observer of the reference input, indicate the expediency of formulating and solving the synthesis problem by interval methods. The transition to interval estimates of the feasible search area makes it possible to obtain a guaranteed solution to a multidimensional synthesis problem with a significant reduction in its computational complexity.