Uncertain dynamical systems. Stability and motion control
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Estimates of the stability of their motion are based on the method of characteristic equations 1 2 Uncertain Dynamical Systems: Stability and Motion Control or its generalizations. Another direction is concerned with the investigation of the dynamics of systems under control, i.
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Here both algebraic and qualitative methods of analysis of the interval stability are used. A generalized view on the role of Riccati equations and many results obtained in this direction are presented in the joint monograph edited by Bittanti, Laub, Willems . Some time ago Boyd et al. This approach is closely connected with the works of Yakubovich  and others. For linear differential inclusions it was developed and presented in the above mentioned monograph by Boyd et al.
Introduction 3 Common in those analyzes is the search of sufficient and sometimes also necessary conditions for the stability or the asymptotic stability of the zero solution of an uncertain system of equations of perturbed motion. One of the effective approaches applied in the analysis of such kind of systems is the method of Lyapunov functions.
Both scalar Lyapunov functions see Corless and Leitmann  and , Gutman , Corless , Leitmann  and , Rotea and Khargonekar , and others and vector Lyapunov functions are used see Chen  and the list of literature to it. Thus, the construction of the theory of stability of systems with uncertain values of their parameters both controlled and free ones is an actual problem of current investigations in nonlinear dynamics and the system theory. Let us deal with two general concepts of stability for systems with uncertain changing parameters.
As against the situation when the parameters of a system are fixed, changing parameters or one parameter causes the occurrence of new equilibrium states of the system. This circumstance does not allow the direct application of the generally used technique of analysis of stability, which was developed for the unique equilibrium state of the system. Here 4 Uncertain Dynamical Systems: Stability and Motion Control some questions of qualitative analysis of the system 1.
The answers to those questions can be found within the framework of the concept of the parametric stability of the system 1. Definition 1. By slight modification of the Definitions 1. In Appendix reader will find the general results of the estimate of the parameter domain and the phase space domain, under which the conclusions about the parametric stability of a dynamic system are correct. If the system 1. Thus, to analyze the system 1.
The noted concept of the parametric stability has been developed and applied to a certain extent. In particular, the problem of the parametric stability of the system 1. The parametric absolute stability of one class of singularly perturbed systems is analyzed in the work by Silva and Dzul  on the basis of the comparison principle for slow and fast variables. Close to this direction are the results of the analysis of stability of systems under nonclassic parametric perturbations, made on the basis of matrix-valued Lyapunov functions see Martynyuk and Miladzhanov .
The stability of solutions of the system 1. From the physical point of view, the case a corresponds to the dynamic properties of the system 1. Note that the investigation of motions of the system 1. Unlike the work by Lakshmikantham and Vatsala , our definition of the set A r makes it possible to consider the systems 1. Later in this monograph the results of the analysis of stability of solutions of some types of simultaneous equations with uncertain values of parameters are made on the basis of both traditional and new approaches.
The objects for analysis in this monograph are some classes of systems of nonlinear and linear equations with uncertain values of parameters in short, uncertain systems. The application of matrix-valued, vector, and scalar Lyapunov functions in combination with the comparison principle brings our generalization into correspondence with the directions a — c above, which are being intensely developed today. The accepted definitions of the stability of motion are given below; they are made specific in separate chapters of this monograph as applied to the considered systems.
Definition 2. Remark 2. We will give some definitions of the stability of motions with respect to a movable invariant set. As required, the general definitions given here will be modified in conformity with the considered class of systems. It is assumed that auxiliary functions can be constructed for nominal systems corresponding to the considered uncertain systems. In the expression 2.
The research of these functions and their application in the stability theory was initiated by Hahn . Following this monograph, we will describe some properties of comparison functions. Below readers will find some definitions applied in this monograph in the analysis of the dynamic behavior of uncertain systems. The matrix-valued function 2. It is supposed that every element vij t, x of the matrix-valued function 2.
Lyapunov discovered a class of auxiliary functions which make it possible to investigate the stability of solutions of simultaneous equations of perturbed motion of quite a general form without its direct integration. Those functions have a number of special properties described in the Definitions 2. The sign definiteness of the Lyapunov function and its full derivative, in view of the system 2.
Classical auxiliary functions are the basis for the direct Lyapunov method. We will give examples of some types of Lyapunov functions. Example 2. The sufficient conditions for stability of the solutions of the uncertain system 2. Theorem 2. Then the set A r is invariant with respect to motions of the system 2.
Proof Let all the conditions of the Theorem 2. Let us prove that the set A r is invariant with respect to the solutions of the uncertain system 2. Suppose this is not so. From the conditions 2 c — e and 3 of the Theorem 2. The inequalities 2. Now prove the stability of the solutions of the system 2.
Suppose that A r is invariant with respect to the solutions of the system 2. Suppose that this is not so. This contradicts the inequality 2. The statement of the Theorem 2. Apply the Theorem 2. Let E0 be the constant energy of the system 2. The conditions a , b indicate that outside the moving surface A r the system 2.
Among them are: fall of a point body with the mass m in a viscous medium with resistance proportionate to velocity; a power grid consisting of one linear constant resistance, a linear constant energy store, and a single external source of energy; processes including chemical changes or compounding processes. Here is an example of such a system. Note that in the case in point the equation 2.
As in the previous case, some parameters in the relevant systems of differential equations may be known uncertainly. We will give several examples of second-order systems. From 2. Proof The invariance of the set A r with respect to the system 2. All its conditions are satisfied when the conditions of the Theorem 2. The uniform asymptotic stability of solutions of the system 2. Let, with all conditions of the Theorem 2. Since in view of the conditions 2 c and 3 a of the Theorem 2. According to the conditions of the Theorem 2. From the inequality 2. But this contradicts the condition 2 a of the Theorem 2.
Therefore the case a is impossible, i. Then from 2. From the condition 2 b of the Theorem 2. Then the set A r is invariant with respect to the motions of the system 2. For the proof we will need the following statement. Lemma 2. According to the condition 4 of the Theorem 2. Consider the case a. According to the condition 3 a of the Theorem 2. Obviously, p t is a locally integrable function on T0.
Now consider the case b. From the condition 3 c of the Theorem 2. This contradicts the condition 2. Taking into account some results of the article by Corless and Leitmann , one can give the following definition. Proof First, prove the invariance of the moving set A r. According to the condition 2 a , b of the Theorem 2. Assume that the set A r is not invariant.
Hence, taking into account 2. The Theorem 2. Corollary 2. Then the solutions of the uncertain system 2. From the conditions 1 and 2 a of the Theorem 2. From the condition 2 a of the Theorem 2. From the estimate 2. Below is a variant of the Theorem 2. This function is positive-definite and satisfies the conditions 2 a , b of the Theorem 2. Its derivative, in view of the equations 2. So, according to the Theorem 2. This is attained by the inverse transformation of the initial dynamic system, which is used in the next chapter for the synthesis of controls of motions of an uncertain system.
Assume that solutions of the equation 2. Assume that under the conditions of the Theorem 2. Taking into account the condition 2 a of the theorem and the relation 2. Now we prove that solutions of the system 2. According to the condition 3 of the Theorem 2. In the system 2. We now prove the following statement. Proposition 2. Then through the nonlinear transformation 2.
Proof From 2. From the relation 2. From the conditions of the Theorem 2. Now prove the asymptotic properties of solutions of the system 2. Then the set B is conditionally invariant with respect to the set A and uniformly asymptotically stable for the solutions of the system 2. Proof Let the set B be not conditionally invariant with respect to the set A for the system 2. In both cases we obtained the contradictions which prove the first part of the statement of the Theorem 2. From the condition 3 a of the Theorem 2.
This proves that the set B is uniformly stable. Now show that under the conditions of the Theorem 2. The contradictions a and b prove the statement of the Theorem 2. Thus, the Theorems 2. Those theorems have wide potential for the further development and applications at the analysis of specific problems of mechanics.
Chapter 3 Stability of Uncertain Controlled Systems In this chapter the construction of controls in the system 3. As follows from the previous chapter, this problem can be reduced to the study of the conditional invariance and the uniform asymptotic stability of the solutions of the system 3. The system 3. Then we will need some assumptions on the systems 3. Assumption 3. Stability of Uncertain Controlled Systems 57 Assumption 3. Now show that the controls 3. Now represent the system 3. Now we will need the following assumptions. The Theorems 2. Theorem 3.
Proof At first consider the dynamics of the uncertain system 3. With this purpose, apply the Lyapunov function V t, x indicated in the Assumption 3. From the inequality 3. When the condition 1 of the Theorem 3. As above, we obtain the inequality DV t, y 3. This inequality, together with the Assumptions 3. The Theorem 3. Prove the following statement. Then the set B is conditionally invariant with respect to the set A, and the motions of the system 3. Proof At first consider the dynamics of the system 3.
Therefore in the domain kxk Stability of Uncertain Controlled Systems 65 Along with the inequalities 3. Taking into account the conditions H2 , the conditions of the Theorem 3. Then, repeating the reasoning of the proof of the Theorem 2. It is assumed that the rigid body is controlled by three bounded forces at three unknown components but with known boundaries of change of the resistance forces.
It is necessary to construct controls under which the motions will be stabilized to the surface of a moving ellipsoid whose surface is an invariant set for the motions of the considered system. I3 I3 3. A special case of the problem 3. Thus, the controls 3. It is assumed that the pair A0 , B is controlled. Let W1 and W2 be some weight matrices characterizing respectively the closed and the output layers of the continual neuron net.
Further in this section, components of the vector-function F h are functions of the form 2.
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We now determine the conditions for the stability of the zero solution of the uncertain system 3. At first, we prove the following statement. Lemma 3. Hence follows the inequality 3. Then the neuron control 3. Proof Let for the nominal system corresponding to 3. Having calculated the full derivative of this 70 Uncertain Dynamical Systems: Stability and Motion Control function along the solutions of the system 3.
Consider the third summand. The control u is considered to be linear with respect to the state vector x t , i. With regard to the system 3. Under the conditions of the Assumption 3. Taking into account the conditions of the Assumption 3. Note the spectral norm of the matrix.
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Let for the equation 3. The following statements should be proved. Remark 3. In other words, 3. Then the system 3. Having estimated the value c using the inequality 3. However, under the condition 3. So the chosen values a and b satisfy the inequality 3.
The inequality 3. Example 3. Assuming that in the system 3. The estimates of uncertainties 3. Chapter 4 Stability of Quasilinear Uncertain Systems Quasilinear systems with uncertain values of parameters are typical models of many events and processes in technical applications. In this chapter we will analyze the problem of the stability of solutions of a quasilinear uncertain system consisting of two subsystems.
It is assumed that the linear approximation of each of the subsystems is reducible to the diagonal form or to the Jordan form. The conditions for the stability of solutions with respect to a moving invariant set are determined on the basis of the results of Chapter 2 in such a form that for their actual verification, the sign-definiteness of special matrices is analyzed.
Along with the matrix-valued Lyapunov function, the vector Lyapunov function is also used in this chapter as a particular case of the matrix-valued function. Several results are illustrated by specific examples. Taking into account the systems of equations 4. In the systems 4.
Therefore the number of different equations in the systems 4. For the system 4. The function 4. For that we introduce some assumptions. Assumption 4. If all the conditions of the Assumptions 4. Then for the full derivative of the function 4. The estimates 4. Theorem 4. The proof is similar to that of the Theorem 2.
Stability of Quasilinear Uncertain Systems 4. As in Section 4. The analysis of the behavior of solutions of the system 4. In the system 4. Along with the system 4. Now consider the motion of the system 4. Let the motion of the system 4. From the estimate 4. In this case the system 4. Then, taking into account 4. From the inequalities 4. Hence the unperturbed motion of the uncertain system 4.
Now analyze the dynamic behavior of solutions of the system 4. Case A. Assume that in the system 4. Case B. Case C. The solution of the system 4.
Lemma 4. Then the solution e t, t0 , x0 of the system 4. Proof For the matrix P satisfying the inequality 4. Taking into account the inequalities 4. Stability of Quasilinear Uncertain Systems 97 Example 4. In this chapter we will discuss some approaches to the solution of the above mentioned problem. They are based on the results of Chapter 2 and use the ideas of hierarchical and vector Lyapunov functions. Let the system 2. As in the case of the system 2. From the general form of the system 5. For example, gi t, x1 ,. The system 5. We now determine the sufficient conditions of the uniform asymptotic stability of solutions of the system 5.
Remark 5. Those approaches, though slightly modified, can be applied in the analysis of the uncertain systems 5. Assume that for each of the subsystems 5. The properties of the motion of the subsystems 5.
Theorem 5. Then the set A r is invariant for the solutions of the system 5. Proof Consider the motion of the system 5. Now show that under the conditions 2 c , 3 a , and 4 a of the Theorem 5. From 5. Since the matrix Q is symmetrical, all its eigenvalues are real. Therefore the inequality 5. A small generalization of the Theorem 5. Proof At first prove the uniform asymptotic stability of the motions of the system 5.
As in the proof of the Theorem 5. Similarly to the inequality 5. Along with the other conditions of the Theorem 5. Now consider the case of stability of motions of the system 5. Under the conditions 2 d of the Theorem 5. This function is constructed on the basis of the two-level decomposition of the system 2. Assume that the system 2. Along with the system 5. The transformation of the system 2.
Assumption 5. Lemma 5. Then the set A r is invariant for solutions of the system 2. Proof Under the conditions of the Theorem 5. Therefore we will give some fragments of the proof. In particular, under the conditions of the Assumption 5. From the condition 2 of the Assumption 5. From the conditions 2 c , e and 3 a , b of the Theorem 5. Proof Apply the scalar approach in the method of vector Lyapunov functions. Under the condition 3 of the Assumption 5. Taking into account the relations 5. Since by the condition of the Theorem 5.
For the system 5. Chapter 6 Interval and Parametric Stability of Uncertain Systems The concept of vector Lyapunov functions is adapted to many dynamic problems of nonlinear systems. In this chapter vector Lyapunov functions are used, both canonical ones and those whose components are quadratic forms. Along with differential inequalities, vector Lyapunov functions present efficient tools for the analysis of stability of uncertain systems. It is known see Tikhonov [a] that in the equations 6. To analyze the behavior of solutions of the system 6. For the full derivatives of the components 6.
We now indicate the conditions under which the solutions of the system 6. Theorem 6. Then for the system 6. Proof From the components 6. Obviously, the function 6. Taking into account 6. For the estimate 6. By the condition 3 a of the Theorem 6. Taking into account the conditions 6. The actual check of these conditions when solving specific problems is made by the method corresponding to the specification of constraints for uncertainties parameters. The development of new efficient procedures of checking those conditions, parallel with the existing ones, is of separate interest.
The system dy e 6. The system 6. Along with the system 6. Assume that in the system 6. Definition 6. The conditions for the interval stability of the system 6. Then the system 6. Proof For the expanded system 6. Consider the full derivatives of the components of the vector Lyapunov function along solutions of the system 6. This completes the proof of the Theorem 6. Proof Consider a system which is an expansion of the system 6.
For the system 6. Let the function f x, p be sufficiently smooth and let the conditions for the existence and uniqueness of the initial problem for the system 6. Represent the system 6. Instead of the system 6. Thus, along with the system 6. Assumption 6. To apply the comparison method for the solution of the problem, see below some auxiliary results.
Lemma 6. The obtained contradiction completes the proof of the Lemma 6. Let us formulate the criterion of Sevastianov-Kotelanskiy see Matrosov  , we will need it for the solution of the main theorem. Proposition 6. Note that later for matrices the Hilbert-Schmidt norm will be used. Estimate the derivatives of the components of the vector Lyapunov function along solutions of the system 6.
It is Uncertain Dynamical Systems: Stability and Motion Control known that the function Vi zi tolerates the following estimate of the time derivative in view of the system 6. Continuing the estimate of 6. Denote , Then, according to the lemma 6. The theorem is proved.
Example 6. Estimate the derivatives of its time components in view of the solutions of the system 6. From the inequality 6. Among systems of this type, which are applied in practice, are fuzzy, hybrid, and impulsive systems. The objective of this chapter is to obtain the conditions for the stability of the motion of impulsive systems with uncertainly known parameters. For this purpose we propose to use the block-diagonal Lyapunov function and the comparison principle. The impulsive system 7. The set of all points of impulsive action upon the continuous component of the system 7. Then the analysis of the strict stability of the system 7.
Assumption 7. Auxiliary functions of the form 7. Particular cases are two-component vector functions; one of their components characterizes the continuous component, the other one characterizes the discrete component. The matrix-values function U1 t, x is constructed on the basis of r stable subsystems of the continuous part of the system 7. Theorem 7. Proof The statement of the Theorem 7. Definition 7. Owing to the condition 2 of the Theorem 7. Under the condition 3 of the Theorem 7.
Case 2. For the system 7. The theory of differential equations implies the following statement. Then the change of the components of the vector function 7. Recall the definition of stability in the sense of Lyapunov, adapted to the uncertain impulsive system 7. From the inequality 7. Lemma 7. Proof Let us show that inequality 7.
Since V x is positive definite, inequality 7. Taking this into account we estimate the second additive in estimate 7. In view of estimate 7. It is easy to see that under condition c of Theorem 7. The following statement holds. Hence it follows that under condition 4 of Theorem 7.
Corollary 7. Then, if condition 4 of Theorem 7. Example 7. Then condition 1 of Theorem 7. From the Theorem 7. For example, Corollary 7. Then the properties of stability of the zero solutions of the comparison system 7. Note that if in the conditions of the Theorem 7.
The system 7. It is obvious that the system 7. The problem of the analysis of dynamic properties of the system 7. For example, for the above mentioned classes of systems it would be interesting to obtain the results described in Chapter 2 for systems of ordinary differential equations with uncertain parameter values.
Chapter 8 Stability of Solutions of Uncertain Dynamic Equations on a Time Scale Dynamic equations on a time scale are adequate models, for the description of continuous- and discrete-time processes occurring in many fields of natural science and technology. In this chapter the reader will find the results of the analysis of the stability of uncertain dynamic equations on the basis of the generalized direct Lyapunov method.
Further information can be found in Bohner and Peterson  with an extensive list of references. A time scale T is an arbitrary nonempty closed subset of a set of real numbers R. Examples of a time scale are: the set R, integers Z, natural numbers N, and nonnegative natural numbers N0. Theorem 8.
Some properties of the integration on T are contained in the following statement. To determine an exponential function on a time scale T the following notions are required. The following properties of the exponential function 8. On the basis of the exponential function 8. Make the following assumptions regarding the system 4. For the system of dynamic equations 8. Remember some definitions. Assume that the elements vij t, x of the function 8. On the basis of the function 8.
Proposition 8. Taking this into account, we represent the estimates a , b from the condition 2 of the Theorem 8. Proof Since the matrices B1 and B2 are positive-definite, we transform the inequalities 2 a , b of the Theorem 8. From the conditions 2 c , d of the Theorem 8. Here, according to the Theorem 8. Taking into account the condition 8. Corollary 8. Proof Under the conditions 2 a , c of the Theorem 8.
The comparison equation for the inequality 8. From the condition 3 of the Theorem 8. In this case from 8. Note that for the system of dynamic equations 8. In some cases it is acceptable to apply simple Lyapunov functions in the analysis of the stability of uncertain dynamic equations of the form 8. Example 8. The uncertain system 8.
Taking into account the properties of exponential functions, transform the estimate 8. Assume that the system 8. Apply the technique of estimates, similar to that used in the Example 8. From the estimate 8. To obtain an estimate of the form 8. Note that equations on time scales, even if uncertainties are not taken into account, have a wider spectrum of dynamic properties as compared both with ordinary differential equations and difference equations.
Here are some examples. In this chapter we propose some development of the direct Liapunov method for the given class of systems of equations in terms of auxiliary matrix valued functions. The chapter is arranged as follows. Section 9. Sections 9. In Section 9. In final Section 9. The class of systems of equations under consideration furtheron designated as F is described basing on the hypotheses below cf.
System F consists of q subsystems of ordinary differential equations with structural uncertainties and r subsystems with structural uncertainties and small parameters at higher derivatives. The order of fast and slow components of the system remains unchanged during all the periods of system F functioning. In this case the time scaling is nonuniform. Therefore systems 9. General purpose of our investigation is to determine conditions under which stability of zero solution of the initial system is implied by stability of some independent degenerated subsystems and stability of the independent subsystems describing boundary layer with allowance for the qualitative properties of interconnections between the subsystems.
Then system 9. In view of results from Martynyuk and Miladzhanov  we introduce some assumptions. Assumption 9. For the function 9. Proof Let all conditions of Assumption 9. Then for function 9. The upper estimate is proved in the same manner. Proposition 9. Then for the expression 9. This estimate allows us to state the following. Theorem 9. Proof Under conditions of Assumption 9. Conditions of Assumption 9.
These conditions are sufficient for uniform asymptotic stability of the equif librium state of system 9. This fact together with the other conditions of the theorem prove the second statement. Remark 9. For system 9. Therefore, by Theorem 9. To study system 9. Proof of Proposition 9. This fact together with the other conditions of Theorem 9. For such choice of the elements of matrix-function 9. So, all conditions of Theorem 9. First, we shall consider the case of nonuniform time scaling.
To this end we need the following assumptions and estimates. The proof is similar to that of Proposition 9. Proof We construct the scalar function v t, x, y, M in the same way as in Section 9. Under the conditions of Assumption 9. The conditions of Assumption 9. These conditions are known to be sufficient for instability of the equilibrium state of system 9. The independent singularly perturbed subsystems corresponding to system 9.
For functions 9. When estimates 9. This theorem is proved in the same manner as Theorem 9. In the matrix-function 9. By Theorem 9. We construct matrix-function 9. For function 9. If estimates 9. We designate the upper boundary of the total derivative of function 9. Note that if the matrix C is negative definite, i. Example 9. For such choice of the elements of matrix 9. Thus, by Theorem 9.
In other words, no matter how accurately one takes into account the forces initiating the phenomenon in question, arbitrarily small perturbations are always left unexplained. The desire to describe this situation adequately generates a need to expand the techniques applicable to the mathematical analysis of the phenomenon in question. Within the framework of the description of phenomena by using ordinary differential equations finite-dimensional ones or equations in Banach spaces , some approaches were proposed which take into account the uncertainness of the values of the system parameters, the fuzziness of systems of differential equations, the inclusion of the derivative of the phase vector into the set of values of the right-hand part of equations of perturbed motion, etc.
All those approaches are designed to take into account the fact that the real motion the stable path is imbedded into the set of other motions paths which occur under the action of unaccounted forces. Hence if the real motion is described by an ordinary differential equation or a system of such equations, then enveloping motions can be described both by ordinary differential equations and by equations with partial derivatives, e.
Under the condition of connectedness of those equations, the obtained set of systems of equations is an example of a hybrid system.