Optimal Control for a Family of Systems in Novel State Derivative Space Form with Experiment in a Double Inverted Pendulum System

Abstract

Optimal control for a family of systems in novel state derivative space form, abbreviated as SDS systems in this study, is proposed. The first step in deriving optimal control laws for SDS systems is to form an augmented cost functional. It turns out that novel differential Lagrange multipliers must be used to adjoin SDS system constraints (namely, the dynamical equations of the control system) to the integrand of the original cost functional which is a function of state derivatives. This not only eases our derivation but also makes our derivation parallel to that for systems in standard state space form. We will show via a real electric circuit that optimal control for a class of descriptor systems with impulse modes can easily be carried out using our design method. It will be shown that linear quadratic regulator (LQR) design for linear time-invariant SDS systems using state derivative feedback can be obtained via an algebraic Riccati equation. Furthermore, this optimal state derivative feedback may also be implemented using an equivalent state feedback. This is useful in real situations when only states but not the state derivatives are available for measurement. The LQR design for a double inverted pendulum system is implemented to illustrate the use of our method.

1. Introduction

Optimal control for systems in standard state space form has long been developed [1–4]. The optimization problem can be stated as follows:

minimize𝐽₀(𝑢)=𝜙𝑥𝑡_𝑓,𝑡_𝑓+^𝑡𝑓_𝑡0𝐹()̇().𝑥,𝑢,𝑡𝑑𝑡subjectto𝑥=𝑓𝑥,𝑢,𝑡 (1a)(1b)

Note that the integrand in the cost functional 𝐽₀(𝑢) is a function of states 𝑥 and controls 𝑢. However, in some applications, the integrand of an appropriate cost functional for the problem at hand is not only a function of states and controls but also a function of the state derivatives 𝑥̇. For instance, to improve people’s comfort and safety when ridding a vehicle or an airplane, acceleration during the course of riding must be taken into account. It is then desirable to minimize the integral of a function of the acceleration during the course. Since acceleration is the derivative of the velocity, which is usually a state variable, we may also consider the cost functional of the form 𝐽₀(𝑢)=𝜙𝑥𝑡_𝑓,𝑡_𝑓+^𝑡𝑓_𝑡0𝐹(𝑥̇,𝑢,𝑡)𝑑𝑡. (2)

Note also that not every control system can be modeled in standard state space form or can easily be handled if it is expressed in standard state space form. For instance, consider the following system: 𝐸𝑥̇=𝐹𝑥+𝑁𝑢. (3) If the coefficient matrix in state derivatives, 𝐸, is singular, then the system cannot be expressed in standard state space form. Such systems are called generalized state space systems [5], descriptor systems [6], singular systems [7], or semistate systems [8]. If 𝐸 is nearly singular, then it is not easy to handle such a system using the standard control design.

Extensive applications of descriptor systems arise in many areas of engineering such as electrical networks [8], aerospace systems [9], smart structures [10], and chemical processes [11]. Descriptor systems also appear in other areas such as the dynamic Leontief model for economic production sectors [12] and biological complex systems [13]. A comprehensive review is available in [14]. In previous studies, descriptor systems are further categorized as impulse-free systems and systems with impulse modes [14]. If the descriptor system has impulsive modes, the integration part of the cost functional may become infinite. In this situation, further investigation of impulse controllability and the impulse mode elimination has to be analyzed in control designs [15]. Therefore, descriptor systems with impulse modes are considered to be difficult in control designs. On the other hand, control designs for impulse-free descriptor systems can be carried out in easier ways. Variational calculus for impulse-free descriptor systems was derived by Jonckheere [16]. Since then, many optimal control algorithms were proposed for impulse-free descriptor systems [14]. Roughly speaking, the available control design algorithms for descriptor systems [15, 17–20] are much more complex than those for the systems in standard state space form. Consequently, there are barricades for engineers without strong mathematical background to apply those sophisticated control algorithms.

If the matrix 𝐹 in (3) is nonsingular, it can be expressed in reciprocal state space (RSS) form [10, 21] as follows: 𝑥=𝐴𝑥̇+𝐵𝑢. (4) Since the system eigenvalues are the reciprocals of the eigenvalues of matrix 𝐴 in (4), the name of reciprocal state space form was given by the first author of this paper. In this form, one can easily carry out control designs such as pole placement [22], eigenstructure assignment [22], and linear quadratic regulator (LQR) designs [22] using state derivative feedback. Therefore, as long as a descriptor system with impulsive mode can be expressed in RSS form, control design can be easily performed by applying state derivative feedback.

Motivated by the analysis above, instead of the systems in standard state space form, a family of control systems in novel state derivative space form, abbreviated as SDS systems, is investigated in this study. The dynamic equation of an SDS system can be described as 𝑥=𝑓(𝑥̇,𝑢,𝑡). (5)

2. Optimality Conditions

The optimal control problem in this study can be stated as follows:

minimize𝐽₀(𝑢)=𝜙𝑥𝑡_𝑓,𝑡_𝑓+^𝑡𝑓_𝑡0𝐹(̇)(̇).𝑥,𝑢,𝑡𝑑𝑡subjectto𝑥=𝑓𝑥,𝑢,𝑡 (6a)(6b)

Further, we assume that the terminal time 𝑡_𝑓 is free and the final states are free but must be constrained to the surface defined by Ψ[𝑥(𝑡_𝑓)]=0. Notice that the integrand of the cost functional may also include the state 𝑥; namely, 𝐹=𝐹(𝑥,𝑥̇,𝑢,𝑡). Simply using the system equation (6b), this function 𝐹 can be reduced to a function that depends only on 𝑥̇, 𝑢, and 𝑡.

To handle this optimization problem, we first form an augmented cost functional as follows: 𝐽=𝜙𝑥𝑡_𝑓,𝑡_𝑓+𝜈^𝑇Ψ𝑥𝑡_𝑓+^𝑡𝑓_𝑡0𝐹(𝑥̇,𝑢,𝑡)+𝜆̇^𝑇(𝑡)𝑓(𝑥̇,𝑢,𝑡)−𝑥𝑑𝑡. (7)

Here, 𝜈 is a vector of usual Lagrange multipliers for the final state constraints. Note that novel differential Lagrange multipliers 𝜆̇, instead of the usual 𝜆, have been used to adjoin SDS system constraints (6b) to the integrand of the original cost functional which is a function of state derivatives. This not only eases our derivation but also makes our derivation parallel to that for systems in standard state space form.

The Hamiltonian is defined by 𝐻𝑥̇,𝑢,𝜆̇,𝑡=𝐹(𝑥̇,𝑢,𝑡)+𝜆̇^𝑇(𝑡)𝑓(𝑥̇,𝑢,𝑡). (8) Substituting (8) to (7), we have 𝐽=𝜙𝑥𝑡_𝑓,𝑡_𝑓+𝜈^𝑇Ψ𝑥𝑡_𝑓+^𝑡𝑓_𝑡0𝐻−𝜆̇^𝑇𝑥𝑑𝑡. (9) Now consider the following small perturbations: 𝑥⟶𝑥+𝛿𝑥,𝑢⟶𝑢+𝛿𝑢,𝑡_𝑓=𝑡_𝑓+𝑑𝑡_𝑓. (10) To minimize (9), the variation of (9) must be zero. That is, 0=Δ𝐽=𝜕𝜙𝜕𝑥+𝜈^𝑇𝜕Ψ𝜕𝑥_{𝑡=𝑡𝑓}𝑑𝑥𝑡_𝑓+𝜕𝜙𝜕𝑡_𝑓𝑑𝑡_𝑓+^𝑡𝑓_𝑡0𝛿𝐻−𝜆̇^𝑇𝑥𝜕𝑑𝑡+𝜕𝑡_𝑓^𝑡𝑓_𝑡0𝐻−𝜆̇^𝑇𝑥𝑑𝑡𝑑𝑡_𝑓. (11) The individual terms of variation in (11) can be obtained as follows: 𝛿𝐻=𝜕𝐻𝜕𝑥̇𝛿𝑥̇+𝜕𝐻̇𝜆𝜕𝑢𝛿𝑢+𝛿^𝑇𝜕𝐻𝜕𝜆̇^𝑇=𝜕𝐻𝜕𝑥̇𝛿𝑥̇+𝜕𝐻̇𝜆𝜕𝑢𝛿𝑢+𝛿^𝑇𝛿−𝜆̇𝑓,_𝑇𝑥=−𝛿𝜆̇^𝑇𝜕𝜆̇^𝑇𝑥𝜕𝜆̇^𝑇−𝜕𝜆̇^𝑇𝑥𝜆̇𝜕𝑥𝛿𝑥=−𝛿^𝑇𝑥−𝜆̇^𝑇𝛿𝑥. (12)

Because of the system constraint (6b), we have −𝛿𝜆̇^𝑇𝑥+𝛿𝜆̇^𝑇𝑓=0. Adding (12) yields 𝛿𝐻−𝜆̇^𝑇𝑥=𝜕𝐻𝜕𝑥̇𝛿𝑥̇+𝜕𝐻̇𝜆𝜕𝑢𝛿𝑢−^𝑇∴𝛿𝑥,_{𝑡𝑓𝑡0}𝛿𝐻−𝜆̇^𝑇𝑥𝑑𝑡=^𝑡𝑓_𝑡0𝜕𝐻𝜕𝑥̇𝛿𝑥̇+𝜕𝐻̇𝜆𝜕𝑢𝛿𝑢−^𝑇𝛿𝑥𝑑𝑡. (13)(14) Integrating by parts for the term −𝜆̇^𝑇𝛿𝑥, we obtain −^𝑡𝑓_𝑡0𝜆̇^𝑇𝛿𝑥𝑑𝑡=−^𝑡𝑓_𝑡0𝜆̇^𝑇𝑑𝑡𝛿𝑥=−^𝑡𝑓_𝑡0𝑑𝜆𝑑𝑡^𝑇𝑑𝑡𝛿𝑥=−^𝑡𝑓_𝑡0𝑑𝜆^𝑇𝛿𝑥=−𝜆^𝑇𝛿𝑥^𝑡𝑓_𝑡0+^𝑡𝑓_𝑡0𝜆^𝑇𝑑(𝛿𝑥)=−𝜆^𝑇𝛿𝑥^𝑡𝑓_𝑡0+^𝑡𝑓_𝑡0𝜆^𝑇𝑑(𝛿𝑥)𝑑𝑡𝑑𝑡=−𝜆^𝑇𝑡_𝑓𝛿𝑥𝑡_𝑓+𝜆^𝑇𝑡₀𝛿𝑥𝑡₀+^𝑡𝑓_𝑡0𝜆^𝑇𝛿𝑥̇𝑑𝑡. (15) Substituting (15) back to (14), we have ^𝑡𝑓_𝑡0𝛿𝐻−𝜆̇^𝑇𝑥𝑑𝑡=^𝑡𝑓_𝑡0𝜕𝐻𝜕𝑥̇+𝜆^𝑇𝛿𝑥̇+𝜕𝐻𝜕𝑢𝛿𝑢𝑑𝑡−𝜆^𝑇𝑡_𝑓𝛿𝑥𝑡_𝑓+𝜆^𝑇𝑡₀𝛿𝑥𝑡₀. (16) It can be shown [23] that 𝑑𝑥𝑡_𝑓=𝛿𝑥𝑡_𝑓+𝑥̇𝑡_𝑓𝑑𝑡_𝑓. (17) Therefore, in (16), we have −𝜆^𝑇𝑡_𝑓𝛿𝑥𝑡_𝑓=−𝜆^𝑇𝑡_𝑓𝑑𝑥𝑡_𝑓+𝜆^𝑇𝑡_𝑓𝑥̇𝑡_𝑓𝑑𝑡_𝑓. (18) In (11), we have 𝜕𝜙𝜕𝑡_𝑓𝑑𝑡_𝑓+𝜕𝜕𝑡_𝑓^𝑡𝑓_𝑡0𝐻−𝜆̇^𝑇𝑥𝑑𝑡𝑑𝑡_𝑓=𝐻+𝜕𝜙𝜕𝑡_𝑓−𝜆̇^𝑇𝑥𝑑𝑡_𝑓. (19) Applying (14)–(19) to (11), one obtains Δ𝐽=𝜕𝜙𝜕𝑥+𝜈^𝑇𝜕Ψ𝜕𝑥−𝜆^𝑇_{𝑡=𝑡𝑓}𝑑𝑥𝑡_𝑓+^𝑡𝑓_𝑡0𝜕𝐻𝜕𝑥̇+𝜆^𝑇𝛿𝑥̇+𝜕𝐻𝜕𝑢𝛿𝑢𝑑𝑡+𝜆^𝑇𝑡₀𝛿𝑥𝑡₀+𝐻+𝜕𝜙𝜕𝑡_𝑓−𝜆̇^𝑇𝑥+𝜆^𝑇𝑥̇𝑑𝑡_𝑓=0. (20) Since the final state 𝑥(𝑡_𝑓) is free, in addition to SDS system constraint in (6b), from (20), necessary conditions of the optimization problem in (6a) and (6b) can be stated as follows: 𝜕𝐻𝜕𝑥̇=−𝜆^𝑇,𝜕𝐻𝜆𝜕𝑢=0,_𝑇𝑡₀𝛿𝑥𝑡₀(),𝜆=0initialconditions_𝑇𝑡_𝑓=𝜕𝜙𝜕𝑥+𝜈^𝑇𝜕Ψ𝜕𝑥_{𝑡=𝑡𝑓}(terminalconditions). (21)(22)(23)(24) From (6b) and (8), the “transversality condition” [23] for 𝑡_𝑓 can be determined via the following equation: 𝐻+𝜕𝜙𝜕𝑡_𝑓−𝜆̇^𝑇𝑥+𝜆^𝑇𝑥̇𝑑𝑡_𝑓=𝐹+𝜆̇^𝑇𝑓+𝜕𝜙𝜕𝑡_𝑓−𝜆̇^𝑇𝑥+𝜆^𝑇𝑥̇𝑑𝑡_𝑓=𝐹+𝜆^𝑇𝑥̇+𝜕𝜙𝜕𝑡_𝑓𝑑𝑡_𝑓=0. (25)

In the following simple example, we wish to illustrate the point that when the system under consideration can be expressed in both standard state space (SSS) form and state derivative space (SDS) form, optimal designs are equivalent.

Example 1. Suppose that we wish to find a control 𝑢 to minimize the following cost functional: 𝐽=𝑡_𝑓+^𝑡𝑓₀𝑢²+𝑥̇𝑑𝑡 (26) subject to 1𝑥=−2̇(̇),̇𝑓(),𝑥+𝑢=𝑓𝑥,𝑢)(inSDSform𝑥=−2𝑥+2𝑢=𝑥,𝑢)(inSSSform (27)(28)𝑥(0)=3, and 𝑡_𝑓 is free.
Let us first solve this problem using the SDS approach. The Hamiltonian is given by 𝐻𝑥̇,𝑢,𝜆̇,𝑡=𝑢²+𝑥̇+𝜆̇−12𝑥̇+𝑢. (29) Necessary conditions (21) and (22) give 𝜕𝐻𝜕𝑥̇1=1−2𝜆̇=−𝜆,𝜕𝐻̇𝜕𝑢=2𝑢+𝜆=0; (30) that is, ̇𝜆̇𝜆−2𝜆−2=0,𝑢=−2. (31) Since the initial condition 𝑥(0)=3 is given, we have 𝛿𝑥(0)=0. Hence, (23) is satisfied. Since Ψ=0 and the 𝜙 is independent of 𝑥, the terminal condition (24) gives 𝜆𝑡_𝑓=𝜕𝜙𝜕𝑥_{𝑡=𝑡𝑓}=0. (32) Since 𝑡_𝑓 is free, the transversality condition (25) now becomes 𝐹+𝜆𝑥̇+𝜕𝜙𝜕𝑡_𝑓=𝑢²+𝑥̇+𝜆𝑥̇+1=𝑢²+(𝜆+1)𝑥̇+1=0. (33) In summary, (27) and (31)–(33) constitute the optimality conditions for the system (27) in SDS form.
For comparison, we then solve this problem again for the system (28) in standard state space form using the standard optimization techniques. The Hamiltonian is given by 𝐻(𝑥̇,𝑢,𝜆,𝑡)=𝑢²+𝑥̇+𝜆()=𝑢−2𝑥+2𝑢₂−2𝑥+2𝑢+𝜆(−2𝑥+2𝑢). (34) Necessary conditions for the optimality [23] are given by 𝜕𝐻̇𝜕𝐻𝜕𝑥=−2−2𝜆=−𝜆,̇𝜕𝑢=2𝑢+2+2𝜆=2𝑢+𝜆=0; (35) that is, ̇̇𝜆𝜆−2𝜆−2=0,𝑢=−2. (36) The terminal condition is given by 𝜆𝑡_𝑓=𝜕𝜙𝜕𝑥_{𝑡=𝑡𝑓}=0. (37) The transversality condition is given by 𝐻+𝜕𝜙𝜕𝑡_𝑓=𝑢²+𝜆()=𝑢−2𝑥+2𝑢−2𝑥+2𝑢+1₂+𝑥̇+𝜆(−2𝑥+2𝑢)+1=𝑢²+𝑥̇+𝜆𝑥̇+1=𝑢²+(𝜆+1)𝑥̇+1=0. (38)
It is not surprising to find that necessary conditions (31)–(33) for the system (27) in SDS form and those necessary conditions (36)–(38) for the system (28) in standard state space form are identical. This means that optimal control design for systems that can be expressed in both SDS and standard state space forms is the same.
Solving (27) and (31)–(33), we have 𝑥(𝑡)=3.081⋅𝑒^−2𝑡−0.081⋅𝑒^2𝑡,𝜆(𝑡)=0.162⋅𝑒^2𝑡𝑢(𝑡)−1,=−0.162⋅𝑒^2𝑡,𝑡_𝑓=0.909. (39)

3. Descriptor Circuit System with Impulse Modes

The following circuit shown in Figure 1 is an example of a descriptor system with impulse mode [8]. It is generally considered to be difficult to carry out the optimal control design for such a system. However, we will find that, simply transforming the dynamic equations of the system into the SDS form in the first place, the optimal control design becomes as straightforward as the usual optimal control design approach used for standard state space systems.

Figure 1 

Descriptor circuit system with impulse mode.

Example 2. Consider the descriptor system of the circuit shown on the left-hand side of Figure 1, and its equivalent alternating current circuit is shown on the right. The dynamic equations for the descriptor system are given as ̇01𝐶0𝑣00𝑥=0110𝑥+𝑢,𝑥=^𝐶𝑖_𝐸=𝑥₁𝑥₂. (40) In the figure, 𝑣_𝐶 is the capacitor voltage, 𝑖_𝐸 is the emitter current, and 𝛼 is the common-base current gain of the bipolar junction transistor. Suppose that we wish to make the changing rates of 𝑣_𝐶 and 𝑖_𝐸 to be as small as possible in the time interval [0,1], that is, keeping the values of 𝑣_𝐶 and 𝑖_𝐸 to be constant as possible. Then the cost functional can be selected as 𝐽=¹₀𝑢²+𝑥̇²₁+𝑥̇²₂𝑑𝑡. (41)
The system may be transformed into the SDS form: ̇0𝑥=00𝐶0𝑥+−1𝑢. (42) The Hamiltonian is given by 𝐻=𝑢²+𝑥̇²₁+𝑥̇²₂+𝜆̇₁(−𝑢)+𝜆̇₂𝐶𝑥̇₁. (43) Necessary conditions (21) and (22) give 𝜕𝐻𝜕𝑥̇₁=2𝑥̇₁+𝐶𝜆̇₂=−𝜆₁,𝜕𝐻𝜕𝑥̇₂=2𝑥̇₂=−𝜆₂,𝜕𝐻̇𝜆𝜕𝑢=2𝑢−₁=0. (44) Thus, the optimal control is given by 𝜆̇𝑢=¹2. (45) We may put (42) and (44) in matrix form: 𝑥𝑋=¹𝑥₂𝜆₁𝜆₂=00−0.50𝑥̇𝐶000−200𝐶0−200¹𝑥̇₂𝜆̇₁𝜆̇₂=𝐴𝑋̇. (46) Note that the matrix 𝐴 is nonsingular for any capacitance 𝐶≠0. For illustrational purpose, we let 𝐶=1. The matrix equation (46) can now be rewritten as 𝑋̇=𝐴⁻¹𝑋=0100000−0.5−20000210𝑋=𝐴𝑋. (47) The general solution of (47) is given by 𝑥𝑋=¹𝑥₂𝜆₁𝜆₂=𝑘₁𝑒0.32240.2535−0.8203−0.3986^0.7862𝑡+𝑘₂𝑒0.3224−0.25350.8203−0.3986^{−0.7862𝑡}+𝑘₃()()()()−0.2535cos1.272𝑡−0.3224sin1.272𝑡−0.3986sin1.272𝑡−0.8203cos1.272𝑡+𝑘₄()()()(),−0.2535sin1.272𝑡0.3224cos1.272𝑡0.3986cos1.272𝑡−0.8203sin1.272𝑡 (48) where 𝑘₁,…,𝑘₄ are constants. The terminal conditions (24) imply that 𝜆₁(1)=𝜆₂(1)=0. These, together with the initial states 𝑥₁(0) and 𝑥₂(0), can be used to determine the constants 𝑘₁,…,𝑘₄. If we let 𝑥₁(0)=1 and 𝑥₂(0)=0, then 𝑘₁=0.6756,𝑘₂𝑘=0.2140,₃=−2.8134,𝑘₄=−0.3629. (49) The optimal control given by (45) can then be determined.

4. LQR Design for SDS Systems

In this section, we consider the LQR problem for linear time-invariant SDS systems. Suppose that the optimization problem can be stated as follows:1minimize𝐽=2^𝑡𝑓_𝑡0𝑥̇^𝑇𝑄𝑥̇+𝑢^𝑇̇𝑅𝑢𝑑𝑡subjectto𝑥=𝐴𝑥+𝐵𝑢. (50a)(50b)

For simplicity, we assume that 𝑄 and 𝑅 are symmetric positive definite matrices. The Hamiltonian is given by 𝐻̇̇=1𝑥,𝑢,𝜆,𝑡2𝑥̇^𝑇𝑄𝑥̇+𝑢^𝑇𝑅𝑢+𝜆̇^𝑇(𝐴𝑥̇+𝐵𝑢). (51) Necessary conditions (21) and (22) imply that 𝜕𝐻𝜕𝑥̇=𝑥̇^𝑇𝑄+𝜆̇^𝑇𝐴=−𝜆^𝑇,𝑢^𝑇𝑅+𝜆̇^𝑇𝐵=0; (52) that is, 𝜆=−𝑄𝑥̇−𝐴^𝑇𝜆̇,𝑢=−𝑅⁻¹𝐵^𝑇𝜆̇. (53) For a linear time-invariant SDS system, one can let 𝜆=𝑆𝑥, (54) where 𝑆 is a constant matrix to be determined. By (53) and (54), we have 𝑢=−𝑅⁻¹𝐵^𝑇𝜆̇=−𝑅⁻¹𝐵^𝑇𝑆𝑥̇. (55) Substituting (55) into (50b), the closed-loop system becomes 𝑥=𝐴𝑥̇−𝐵𝑅⁻¹𝐵^𝑇𝑥̇=𝐴−𝐵𝑅⁻¹𝐵^𝑇𝑆𝑥̇. (56) By (53), (54), and (56), we have 𝑆𝐴−𝐵𝑅⁻¹𝐵^𝑇𝑆𝑥̇=𝑆𝑥=𝜆=−𝑄𝑥̇−𝐴^𝑇𝜆̇=−𝑄𝑥̇−𝐴^𝑇𝑆𝑥̇; (57) that is, 𝑆𝐴+𝐴^𝑇𝑆−𝑆𝐵𝑅⁻¹𝐵^𝑇𝑆+𝑄𝑥̇=0. (58) Since this is true for any 𝑥̇, we obtain the following Riccati equation for SDS systems: 𝑆𝐴+𝐴^𝑇𝑆−𝑆𝐵𝑅⁻¹𝐵^𝑇𝑆+𝑄=0. (59) By (55), the optimal state derivative feedback control is given by 𝑢=−𝐾𝑥̇,𝐾=𝑅⁻¹𝐵^𝑇𝑆, (60) and the closed-loop system becomes 𝑥=(𝐴−𝐵𝐾)𝑥̇. (61)

The optimal state derivative feedback control law 𝑢=−𝐾𝑥̇ may also be implemented using an equivalent state feedback 𝑢=−𝐾𝑥. This is useful in real situations when only states but not the state derivatives are available for measurement. This can be proved as follows. The closed-loop system after applying 𝑢=−𝐾𝑥 becomes 𝑥=𝐴𝑥̇−𝐵𝐾𝑥, (62) implying 𝐼+𝐵𝐾𝑥=𝐴𝑥̇, (63) or 𝑥=𝐼+𝐵𝐾⁻¹𝐴𝑥̇. (64) In view of (61), we set 𝐼+𝐵𝐾⁻¹𝐴=(𝐴−𝐵𝐾). (65) Then, we have 𝐴=𝐼+𝐵𝐾(𝐴−𝐵𝐾)=𝐴+𝐵𝐾𝐴−𝐵𝐾−𝐵𝐾𝐵𝐾, (66) or 𝐵𝐾𝐴−𝐾−𝐾𝐵𝐾=0. (67) Now let 𝐾𝐴−𝐾−𝐾𝐵𝐾=0, (68) implying 𝐾=𝐾(𝐴−𝐵𝐾)⁻¹. (69)

5. Controlling a Double Inverted Pendulum

Consider the double inverted pendulum system shown in Figure 2 [24, 25], where 𝜃₁ is the angle of inclination of the first arm and 𝜃₂ is the angle of inclination of the second arm relative to the first arm. This experiment facility is controlled by Simulink interface [24]. As shown in the figure, there are two subcontrol systems in this double inverted pendulum.

Figure 2 

Double inverted pendulum.

Our goal is to keep the first arm in a vertical position pointing downward and second arm in a vertical position but pointing upward. This means that we wish that 𝜃₁=−90^∘,𝜃₂=180^∘,𝜃̇₁=0,𝜃̇₂=0. (70) For this system, an initial control is usually needed so that 𝜃₂ approximately lies between 170^∘ and 190^∘ [25, 26]. An appropriate linearized model, accurate enough for 𝜃₂∈[170^∘,190^∘], is given by [24]: ̇00𝑥=00100001−15.8868.0562−0.0405082.08395.9987−0.02030𝑥+2.16491.8943𝑢, (71) where 𝑥^𝑇=𝜃₁𝜃₂𝜃̇₁𝜃̇₂ and 𝑢 is the input voltage. Suppose that we wish to minimize the following cost functional: 1𝐽=2^𝑡𝑓_𝑡0𝑥̇^𝑇𝑄𝑥̇+𝑢^𝑇𝑅𝑢𝑑𝑡, (72) where 𝑄=𝑅=𝐼. The controllable system (71) can be converted in SDS form: ̇𝑥̇+00𝑥=𝐴𝑥+𝐵𝑢=−0.00170−0.04390.00370.001700.03750.0073100001000.0881−0.0950𝑢. (73) Using the method proposed in the last section, the optimal state derivative feedback control law (60) is given by 𝑢=−𝐾𝑥̇, where 𝐾=−10.9858−16.5454−0.1404−19751. (74) An equivalent state feedback control law is given by 𝑢=−𝐾𝑥, where 𝐾=52.500962.62983.59225.4327. (75) We will use the state feedback control law in our real implementation through the Simulink control interface given in [24]. Typical simulated trajectories for 𝜃₁ and 𝜃₂ are shown in Figures 3 and 4, respectively. It is seen that the control goal is achieved. The full-length video taken during the experiment is available in [27].

Figure 3 

Trajectory of 𝜃1.

Figure 4 

Trajectory of 𝜃2.

6. Conclusion

Optimal control for SDS systems has been investigated in this paper. Necessary conditions for optimality were derived by the use of novel differential Lagrange multipliers. In the past, optimal control design for descriptor systems with impulse modes is not an easy task. We have shown via a real electric circuit that optimal control for a class of descriptor systems with impulse modes can easily be carried out using our design method. The optimal LQR design for linear time-invariant SDS systems using state derivative feedback can be obtained via an algebraic Riccati equation. Moreover, this optimal state derivative feedback may also be implemented using an equivalent state feedback. This is useful in real situations when only states are available for measurement. Note that our derivation and design method parallel those for systems in standard state space form. The LQR design for a double inverted pendulum system has been implemented to illustrate the use of our method.

Acknowledgment

The research reported here was supported by the National Science Council, Taiwan, under Grants NSC100-2221-E-214-017 and NSC101-2221-E-214-042.